diff --git a/pandas/core/sorting.py b/pandas/core/sorting.py
index 930704e6f62f4..3349a13d7f0bc 100644
--- a/pandas/core/sorting.py
+++ b/pandas/core/sorting.py
@@ -57,7 +57,7 @@
 
 def get_indexer_indexer(
     target: Index,
-    level: Level | list[Level] | None,
+    level: Level | list[Level] | None,  # can level actually be a list here?
     ascending: list[bool] | bool,
     kind: SortKind,
     na_position: NaPosition,
@@ -531,7 +531,6 @@ def _ensure_key_mapped_multiindex(
             level_iter = [level]
         else:
             level_iter = level
-
         sort_levels: range | set = {index._get_level_number(lev) for lev in level_iter}
     else:
         sort_levels = range(index.nlevels)
@@ -581,7 +580,10 @@ def ensure_key_mapped(
         if isinstance(
             values, Index
         ):  # convert to a new Index subclass, not necessarily the same
-            result = Index(result, tupleize_cols=False)
+            # preserve the original name when creating the new Index
+            result = Index(
+                result, tupleize_cols=False, name=getattr(values, "name", None)
+            )
         else:
             # try to revert to original type otherwise
             type_of_values = type(values)
diff --git a/pandas/tests/frame/methods/test_sort_index.py b/pandas/tests/frame/methods/test_sort_index.py
index 1a631e760208a..1ee7d3a91f41b 100644
--- a/pandas/tests/frame/methods/test_sort_index.py
+++ b/pandas/tests/frame/methods/test_sort_index.py
@@ -1,3 +1,4 @@
+from natsort import index_natsorted  # should we import this? or change test?
 import numpy as np
 import pytest
 
@@ -943,6 +944,95 @@ def test_sort_index_multiindex_sort_remaining(self, ascending):
 
         tm.assert_frame_equal(result, expected)
 
+    def test_sort_index_multiindex_by_level_name(self):
+        # GH#62361
+
+        df = DataFrame(
+            [[1, 2], [3, 4]],
+            columns=MultiIndex.from_product(
+                [["a"], ["top10", "top2"]], names=("A", "B")
+            ),
+        )
+
+        expected = DataFrame(
+            [[2, 1], [4, 3]],
+            columns=MultiIndex.from_product(
+                [["a"], ["top2", "top10"]], names=("A", "B")
+            ),
+        )
+
+        sorted_df = df.sort_index(
+            axis=1, level="B", key=lambda x: np.argsort(index_natsorted(x))
+        )
+        tm.assert_frame_equal(sorted_df, expected)
+
+    def test_sort_index_multiindex_by_level_name_2(self):
+        # GH#62361
+
+        df = DataFrame(
+            [[1, 2, 3], [4, 5, 6]],
+            columns=MultiIndex.from_tuples(
+                [("a10", "b12"), ("a2", "b17"), ("a2", "b4")], names=("A", "B")
+            ),
+        )
+
+        expected_A = DataFrame(
+            [[2, 3, 1], [5, 6, 4]],
+            columns=MultiIndex.from_tuples(
+                [("a2", "b17"), ("a2", "b4"), ("a10", "b12")], names=("A", "B")
+            ),
+        )
+        expected_B = DataFrame(
+            [[3, 1, 2], [6, 4, 5]],
+            columns=MultiIndex.from_tuples(
+                [("a2", "b4"), ("a10", "b12"), ("a2", "b17")], names=("A", "B")
+            ),
+        )
+
+        sorted_df = df.sort_index(
+            axis=1, level=0, key=lambda x: np.argsort(index_natsorted(x))
+        )
+        tm.assert_frame_equal(sorted_df, expected_A)
+
+        sorted_df = df.sort_index(
+            axis=1, level="A", key=lambda x: np.argsort(index_natsorted(x))
+        )
+        tm.assert_frame_equal(sorted_df, expected_A)
+
+        sorted_df = df.sort_index(
+            axis=1, level=1, key=lambda x: np.argsort(index_natsorted(x))
+        )
+        tm.assert_frame_equal(sorted_df, expected_B)
+
+        sorted_df = df.sort_index(
+            axis=1, level="B", key=lambda x: np.argsort(index_natsorted(x))
+        )
+        tm.assert_frame_equal(sorted_df, expected_B)
+
+        # actually, only 1 element of list matters for sorting (2nd is ignored)
+        sorted_df = df.sort_index(
+            axis=1, level=[0, 1], key=lambda x: np.argsort(index_natsorted(x))
+        )
+        tm.assert_frame_equal(sorted_df, expected_A)
+
+        # sorted_df = df.sort_index(
+        #     axis=1, level=[1, 0], key=lambda x: np.argsort(index_natsorted(x))
+        # )
+        # tm.assert_frame_equal(sorted_df, expected_B)
+
+        # sorted_df = df.sort_index(
+        #     axis=1, level=[1, "A"], key=lambda x: np.argsort(index_natsorted(x))
+        # )
+        # tm.assert_frame_equal(sorted_df, expected_B)
+
+        # # repetition does not matter
+        # sorted_df = df.sort_index(
+        #     axis=1,
+        #     level=["A", "B", 0, 1, "B"],
+        #     key=lambda x: np.argsort(index_natsorted(x)),
+        # )
+        # tm.assert_frame_equal(sorted_df, expected_A)
+
 
 def test_sort_index_with_sliced_multiindex():
     # GH 55379