diff --git a/datafusion/physical-expr/src/equivalence/ordering.rs b/datafusion/physical-expr/src/equivalence/ordering.rs
index 24e2fc7dbaf5..22696f1fa527 100644
--- a/datafusion/physical-expr/src/equivalence/ordering.rs
+++ b/datafusion/physical-expr/src/equivalence/ordering.rs
@@ -39,7 +39,7 @@ use arrow_schema::SortOptions;
 /// ordering. In this case, we say that these orderings are equivalent.
 #[derive(Debug, Clone, Eq, PartialEq, Hash, Default)]
 pub struct OrderingEquivalenceClass {
-    pub orderings: Vec<LexOrdering>,
+    orderings: Vec<LexOrdering>,
 }
 
 impl OrderingEquivalenceClass {
@@ -53,13 +53,20 @@ impl OrderingEquivalenceClass {
         self.orderings.clear();
     }
 
-    /// Creates new ordering equivalence class from the given orderings.
+    /// Creates new ordering equivalence class from the given orderings
+    ///
+    /// Any redundant entries are removed, as described on [`Self::remove_redundant_entries`].
     pub fn new(orderings: Vec<LexOrdering>) -> Self {
         let mut result = Self { orderings };
         result.remove_redundant_entries();
         result
     }
 
+    /// Converts this OrderingEquivalenceClass to a vector of orderings.
+    pub fn into_inner(self) -> Vec<LexOrdering> {
+        self.orderings
+    }
+
     /// Checks whether `ordering` is a member of this equivalence class.
     pub fn contains(&self, ordering: &LexOrdering) -> bool {
         self.orderings.contains(ordering)
@@ -67,10 +74,12 @@ impl OrderingEquivalenceClass {
 
     /// Adds `ordering` to this equivalence class.
     #[allow(dead_code)]
+    #[deprecated(
+        since = "45.0.0",
+        note = "use OrderingEquivalenceClass::add_new_ordering instead"
+    )]
     fn push(&mut self, ordering: LexOrdering) {
-        self.orderings.push(ordering);
-        // Make sure that there are no redundant orderings:
-        self.remove_redundant_entries();
+        self.add_new_ordering(ordering)
     }
 
     /// Checks whether this ordering equivalence class is empty.
@@ -79,6 +88,9 @@ impl OrderingEquivalenceClass {
     }
 
     /// Returns an iterator over the equivalent orderings in this class.
+    ///
+    /// Note this class also implements [`IntoIterator`] to return an iterator
+    /// over owned [`LexOrdering`]s.
     pub fn iter(&self) -> impl Iterator<Item = &LexOrdering> {
         self.orderings.iter()
     }
@@ -95,7 +107,7 @@ impl OrderingEquivalenceClass {
         self.remove_redundant_entries();
     }
 
-    /// Adds new orderings into this ordering equivalence class.
+    /// Adds new orderings into this ordering equivalence class
     pub fn add_new_orderings(
         &mut self,
         orderings: impl IntoIterator<Item = LexOrdering>,
@@ -110,9 +122,10 @@ impl OrderingEquivalenceClass {
         self.add_new_orderings([ordering]);
     }
 
-    /// Removes redundant orderings from this equivalence class. For instance,
-    /// if we already have the ordering `[a ASC, b ASC, c DESC]`, then there is
-    /// no need to keep ordering `[a ASC, b ASC]` in the state.
+    /// Removes redundant orderings from this equivalence class.
+    ///
+    /// For instance, if we already have the ordering `[a ASC, b ASC, c DESC]`,
+    /// then there is no need to keep ordering `[a ASC, b ASC]` in the state.
     fn remove_redundant_entries(&mut self) {
         let mut work = true;
         while work {
@@ -198,6 +211,7 @@ impl OrderingEquivalenceClass {
     }
 }
 
+/// Convert the `OrderingEquivalenceClass` into an iterator of LexOrderings
 impl IntoIterator for OrderingEquivalenceClass {
     type Item = LexOrdering;
     type IntoIter = IntoIter<LexOrdering>;
@@ -293,13 +307,15 @@ mod tests {
         // finer ordering satisfies, crude ordering should return true
         let mut eq_properties_finer =
             EquivalenceProperties::new(Arc::clone(&input_schema));
-        eq_properties_finer.oeq_class.push(finer.clone());
+        eq_properties_finer
+            .oeq_class
+            .add_new_ordering(finer.clone());
         assert!(eq_properties_finer.ordering_satisfy(crude.as_ref()));
 
         // Crude ordering doesn't satisfy finer ordering. should return false
         let mut eq_properties_crude =
             EquivalenceProperties::new(Arc::clone(&input_schema));
-        eq_properties_crude.oeq_class.push(crude);
+        eq_properties_crude.oeq_class.add_new_ordering(crude);
         assert!(!eq_properties_crude.ordering_satisfy(finer.as_ref()));
         Ok(())
     }
diff --git a/datafusion/physical-expr/src/equivalence/projection.rs b/datafusion/physical-expr/src/equivalence/projection.rs
index 25a05a2a5918..681484fd6bff 100644
--- a/datafusion/physical-expr/src/equivalence/projection.rs
+++ b/datafusion/physical-expr/src/equivalence/projection.rs
@@ -636,7 +636,7 @@ mod tests {
 
             let err_msg = format!(
                 "test_idx: {:?}, actual: {:?}, expected: {:?}, projection_mapping: {:?}",
-                idx, orderings.orderings, expected, projection_mapping
+                idx, orderings, expected, projection_mapping
             );
 
             assert_eq!(orderings.len(), expected.len(), "{}", err_msg);
@@ -825,7 +825,7 @@ mod tests {
 
             let err_msg = format!(
                 "test idx: {:?}, actual: {:?}, expected: {:?}, projection_mapping: {:?}",
-                idx, orderings.orderings, expected, projection_mapping
+                idx, orderings, expected, projection_mapping
             );
 
             assert_eq!(orderings.len(), expected.len(), "{}", err_msg);
@@ -971,7 +971,7 @@ mod tests {
 
             let err_msg = format!(
                 "actual: {:?}, expected: {:?}, projection_mapping: {:?}",
-                orderings.orderings, expected, projection_mapping
+                orderings, expected, projection_mapping
             );
 
             assert_eq!(orderings.len(), expected.len(), "{}", err_msg);
diff --git a/datafusion/physical-expr/src/equivalence/properties.rs b/datafusion/physical-expr/src/equivalence/properties.rs
index c3d458103285..5c0c1e18b2df 100755
--- a/datafusion/physical-expr/src/equivalence/properties.rs
+++ b/datafusion/physical-expr/src/equivalence/properties.rs
@@ -431,8 +431,8 @@ impl EquivalenceProperties {
         let mut new_orderings = vec![filtered_exprs.clone()];
 
         // Preserve valid suffixes from existing orderings
-        let orderings = mem::take(&mut self.oeq_class.orderings);
-        for existing in orderings {
+        let oeq_class = mem::take(&mut self.oeq_class);
+        for existing in oeq_class {
             if self.is_prefix_of(&filtered_exprs, &existing) {
                 let mut extended = filtered_exprs.clone();
                 extended.extend(existing.into_iter().skip(filtered_exprs.len()));
@@ -711,8 +711,8 @@ impl EquivalenceProperties {
     /// Since it would cause bug in dependency constructions, we should substitute the input order in order to get correct
     /// dependency map, happen in issue 8838: <https://github.com/apache/datafusion/issues/8838>
     pub fn substitute_oeq_class(&mut self, mapping: &ProjectionMapping) -> Result<()> {
-        let orderings = &self.oeq_class.orderings;
-        let new_order = orderings
+        let new_order = self
+            .oeq_class
             .iter()
             .map(|order| self.substitute_ordering_component(mapping, order))
             .collect::<Result<Vec<_>>>()?;
@@ -1220,7 +1220,7 @@ impl EquivalenceProperties {
 
         // Rewrite orderings according to new schema:
         let mut new_orderings = vec![];
-        for ordering in self.oeq_class.orderings {
+        for ordering in self.oeq_class {
             let new_ordering = ordering
                 .inner
                 .into_iter()
@@ -2009,16 +2009,8 @@ fn calculate_union_binary(
     // Next, calculate valid orderings for the union by searching for prefixes
     // in both sides.
     let mut orderings = UnionEquivalentOrderingBuilder::new();
-    orderings.add_satisfied_orderings(
-        lhs.normalized_oeq_class().orderings,
-        lhs.constants(),
-        &rhs,
-    );
-    orderings.add_satisfied_orderings(
-        rhs.normalized_oeq_class().orderings,
-        rhs.constants(),
-        &lhs,
-    );
+    orderings.add_satisfied_orderings(lhs.normalized_oeq_class(), lhs.constants(), &rhs);
+    orderings.add_satisfied_orderings(rhs.normalized_oeq_class(), rhs.constants(), &lhs);
     let orderings = orderings.build();
 
     let mut eq_properties =
@@ -2157,7 +2149,7 @@ impl UnionEquivalentOrderingBuilder {
 
         // for each equivalent ordering in properties, try and augment
         // `ordering` it with the constants to match
-        for existing_ordering in &properties.oeq_class.orderings {
+        for existing_ordering in properties.oeq_class.iter() {
             if let Some(augmented_ordering) = self.augment_ordering(
                 ordering,
                 constants,
@@ -2438,17 +2430,12 @@ mod tests {
                 Some(JoinSide::Left),
                 &[],
             );
-            let orderings = &join_eq.oeq_class.orderings;
-            let err_msg = format!("expected: {:?}, actual:{:?}", expected, orderings);
-            assert_eq!(
-                join_eq.oeq_class.orderings.len(),
-                expected.len(),
-                "{}",
-                err_msg
-            );
-            for ordering in orderings {
+            let err_msg =
+                format!("expected: {:?}, actual:{:?}", expected, &join_eq.oeq_class);
+            assert_eq!(join_eq.oeq_class.len(), expected.len(), "{}", err_msg);
+            for ordering in join_eq.oeq_class {
                 assert!(
-                    expected.contains(ordering),
+                    expected.contains(&ordering),
                     "{}, ordering: {:?}",
                     err_msg,
                     ordering
@@ -3767,8 +3754,8 @@ mod tests {
 
         // Check whether orderings are same.
         let lhs_orderings = lhs.oeq_class();
-        let rhs_orderings = &rhs.oeq_class.orderings;
-        for rhs_ordering in rhs_orderings {
+        let rhs_orderings = rhs.oeq_class();
+        for rhs_ordering in rhs_orderings.iter() {
             assert!(
                 lhs_orderings.contains(rhs_ordering),
                 "{err_msg}\nlhs: {lhs}\nrhs: {rhs}"
@@ -3844,7 +3831,7 @@ mod tests {
         // Add equality condition c = concat(a, b)
         eq_properties.add_equal_conditions(&col_c, &a_concat_b)?;
 
-        let orderings = eq_properties.oeq_class().orderings.clone();
+        let orderings = eq_properties.oeq_class();
 
         let expected_ordering1 =
             LexOrdering::from(vec![
@@ -3895,7 +3882,7 @@ mod tests {
         // Add equality condition c = a * b
         eq_properties.add_equal_conditions(&col_c, &a_times_b)?;
 
-        let orderings = eq_properties.oeq_class().orderings.clone();
+        let orderings = eq_properties.oeq_class();
 
         // The ordering should remain unchanged since multiplication is not lex-monotonic
         assert_eq!(orderings.len(), 1);
@@ -3935,7 +3922,7 @@ mod tests {
         // Add equality condition c = concat(a, b)
         eq_properties.add_equal_conditions(&col_c, &a_concat_b)?;
 
-        let orderings = eq_properties.oeq_class().orderings.clone();
+        let orderings = eq_properties.oeq_class();
 
         let expected_ordering1 = LexOrdering::from(vec![PhysicalSortExpr::new_default(
             Arc::clone(&a_concat_b),
@@ -3979,8 +3966,9 @@ mod tests {
 
         // Should only contain b since a is constant
         assert_eq!(result.oeq_class().len(), 1);
-        assert_eq!(result.oeq_class().orderings[0].len(), 1);
-        assert!(result.oeq_class().orderings[0][0].expr.eq(&col_b));
+        let ordering = result.oeq_class().iter().next().unwrap();
+        assert_eq!(ordering.len(), 1);
+        assert!(ordering[0].expr.eq(&col_b));
 
         Ok(())
     }
@@ -4026,13 +4014,14 @@ mod tests {
 
         // Should only contain [a ASC, b DESC, c ASC]
         assert_eq!(result.oeq_class().len(), 1);
-        assert_eq!(result.oeq_class().orderings[0].len(), 3);
-        assert!(result.oeq_class().orderings[0][0].expr.eq(&col_a));
-        assert!(result.oeq_class().orderings[0][0].options.eq(&asc));
-        assert!(result.oeq_class().orderings[0][1].expr.eq(&col_b));
-        assert!(result.oeq_class().orderings[0][1].options.eq(&desc));
-        assert!(result.oeq_class().orderings[0][2].expr.eq(&col_c));
-        assert!(result.oeq_class().orderings[0][2].options.eq(&asc));
+        let ordering = result.oeq_class().iter().next().unwrap();
+        assert_eq!(ordering.len(), 3);
+        assert!(ordering[0].expr.eq(&col_a));
+        assert!(ordering[0].options.eq(&asc));
+        assert!(ordering[1].expr.eq(&col_b));
+        assert!(ordering[1].options.eq(&desc));
+        assert!(ordering[2].expr.eq(&col_c));
+        assert!(ordering[2].options.eq(&asc));
 
         Ok(())
     }
@@ -4075,11 +4064,12 @@ mod tests {
         assert_eq!(result.oeq_class().len(), 1);
 
         // Verify orderings
-        assert_eq!(result.oeq_class().orderings[0].len(), 2);
-        assert!(result.oeq_class().orderings[0][0].expr.eq(&col_b));
-        assert!(result.oeq_class().orderings[0][0].options.eq(&asc));
-        assert!(result.oeq_class().orderings[0][1].expr.eq(&col_c));
-        assert!(result.oeq_class().orderings[0][1].options.eq(&asc));
+        let ordering = result.oeq_class().iter().next().unwrap();
+        assert_eq!(ordering.len(), 2);
+        assert!(ordering[0].expr.eq(&col_b));
+        assert!(ordering[0].options.eq(&asc));
+        assert!(ordering[1].expr.eq(&col_c));
+        assert!(ordering[1].options.eq(&asc));
 
         Ok(())
     }
@@ -4120,7 +4110,8 @@ mod tests {
 
         // Should only contain the new ordering since options don't match
         assert_eq!(result.oeq_class().len(), 1);
-        assert_eq!(result.oeq_class().orderings[0], new_order);
+        let ordering = result.oeq_class().iter().next().unwrap();
+        assert_eq!(ordering, &new_order);
 
         Ok(())
     }
@@ -4178,7 +4169,7 @@ mod tests {
 
         // Should preserve the original [d ASC, a ASC] ordering
         assert_eq!(result.oeq_class().len(), 1);
-        let ordering = &result.oeq_class().orderings[0];
+        let ordering = result.oeq_class().iter().next().unwrap();
         assert_eq!(ordering.len(), 2);
 
         // First expression should be either b or d (they're equivalent)
diff --git a/datafusion/physical-plan/src/memory.rs b/datafusion/physical-plan/src/memory.rs
index 521008ce9b02..ee8f9e1cbfea 100644
--- a/datafusion/physical-plan/src/memory.rs
+++ b/datafusion/physical-plan/src/memory.rs
@@ -261,7 +261,7 @@ impl MemoryExec {
             sort_information = base_eqp
                 .project(&projection_mapping, self.schema())
                 .oeq_class
-                .orderings;
+                .into_inner();
         }
 
         self.sort_information = sort_information;
diff --git a/datafusion/physical-plan/src/union.rs b/datafusion/physical-plan/src/union.rs
index 6e768a3d87bc..cedfcee6fabd 100644
--- a/datafusion/physical-plan/src/union.rs
+++ b/datafusion/physical-plan/src/union.rs
@@ -843,9 +843,9 @@ mod tests {
     ) {
         // Check whether orderings are same.
         let lhs_orderings = lhs.oeq_class();
-        let rhs_orderings = &rhs.oeq_class.orderings;
+        let rhs_orderings = &rhs.oeq_class;
         assert_eq!(lhs_orderings.len(), rhs_orderings.len(), "{}", err_msg);
-        for rhs_ordering in rhs_orderings {
+        for rhs_ordering in rhs_orderings.iter() {
             assert!(lhs_orderings.contains(rhs_ordering), "{}", err_msg);
         }
     }