diff --git a/pandas/core/frame.py b/pandas/core/frame.py
index c7f0a7ced7576..eb3365d4f8410 100644
--- a/pandas/core/frame.py
+++ b/pandas/core/frame.py
@@ -7407,7 +7407,7 @@ def swaplevel(self, i: Axis = -2, j: Axis = -1, axis: Axis = 0) -> DataFrame:
             result.columns = result.columns.swaplevel(i, j)
         return result
 
-    def reorder_levels(self, order: Sequence[Axis], axis: Axis = 0) -> DataFrame:
+    def reorder_levels(self, order: Sequence[int | str], axis: Axis = 0) -> DataFrame:
         """
         Rearrange index levels using input order. May not drop or duplicate levels.
 
@@ -7452,7 +7452,7 @@ class      diet
         if not isinstance(self._get_axis(axis), MultiIndex):  # pragma: no cover
             raise TypeError("Can only reorder levels on a hierarchical axis.")
 
-        result = self.copy()
+        result = self.copy(deep=None)
 
         if axis == 0:
             assert isinstance(result.index, MultiIndex)
diff --git a/pandas/tests/copy_view/test_methods.py b/pandas/tests/copy_view/test_methods.py
index 6707f1411cbc7..8015eb93988c9 100644
--- a/pandas/tests/copy_view/test_methods.py
+++ b/pandas/tests/copy_view/test_methods.py
@@ -3,6 +3,7 @@
 
 from pandas import (
     DataFrame,
+    MultiIndex,
     Series,
 )
 import pandas._testing as tm
@@ -293,7 +294,25 @@ def test_assign(using_copy_on_write):
     else:
         assert not np.shares_memory(get_array(df2, "a"), get_array(df, "a"))
 
-    # modify df2 to trigger CoW for that block
+    df2.iloc[0, 0] = 0
+    if using_copy_on_write:
+        assert not np.shares_memory(get_array(df2, "a"), get_array(df, "a"))
+    tm.assert_frame_equal(df, df_orig)
+
+
+def test_reorder_levels(using_copy_on_write):
+    index = MultiIndex.from_tuples(
+        [(1, 1), (1, 2), (2, 1), (2, 2)], names=["one", "two"]
+    )
+    df = DataFrame({"a": [1, 2, 3, 4]}, index=index)
+    df_orig = df.copy()
+    df2 = df.reorder_levels(order=["two", "one"])
+
+    if using_copy_on_write:
+        assert np.shares_memory(get_array(df2, "a"), get_array(df, "a"))
+    else:
+        assert not np.shares_memory(get_array(df2, "a"), get_array(df, "a"))
+
     df2.iloc[0, 0] = 0
     if using_copy_on_write:
         assert not np.shares_memory(get_array(df2, "a"), get_array(df, "a"))