diff --git a/doc/source/whatsnew/v1.5.0.rst b/doc/source/whatsnew/v1.5.0.rst
index a5cb716317689..d8b8f620a1d62 100644
--- a/doc/source/whatsnew/v1.5.0.rst
+++ b/doc/source/whatsnew/v1.5.0.rst
@@ -963,7 +963,7 @@ Other
 
 .. ***DO NOT USE THIS SECTION***
 
--
+- Bug in :func:`.assert_index_equal` with ``names=True`` and ``check_order=False`` not checking names (:issue:`47328`)
 -
 
 .. ---------------------------------------------------------------------------
diff --git a/pandas/_testing/asserters.py b/pandas/_testing/asserters.py
index 2029a7665d57a..90ee600c1967d 100644
--- a/pandas/_testing/asserters.py
+++ b/pandas/_testing/asserters.py
@@ -45,10 +45,7 @@
     Series,
     TimedeltaIndex,
 )
-from pandas.core.algorithms import (
-    safe_sort,
-    take_nd,
-)
+from pandas.core.algorithms import take_nd
 from pandas.core.arrays import (
     DatetimeArray,
     ExtensionArray,
@@ -58,6 +55,7 @@
 )
 from pandas.core.arrays.datetimelike import DatetimeLikeArrayMixin
 from pandas.core.arrays.string_ import StringDtype
+from pandas.core.indexes.api import safe_sort_index
 
 from pandas.io.formats.printing import pprint_thing
 
@@ -367,8 +365,8 @@ def _get_ilevel_values(index, level):
 
     # If order doesn't matter then sort the index entries
     if not check_order:
-        left = Index(safe_sort(left), dtype=left.dtype)
-        right = Index(safe_sort(right), dtype=right.dtype)
+        left = safe_sort_index(left)
+        right = safe_sort_index(right)
 
     # MultiIndex special comparison for little-friendly error messages
     if left.nlevels > 1:
diff --git a/pandas/core/indexes/api.py b/pandas/core/indexes/api.py
index 1e740132e3464..634187a317baf 100644
--- a/pandas/core/indexes/api.py
+++ b/pandas/core/indexes/api.py
@@ -70,6 +70,7 @@
     "get_unanimous_names",
     "all_indexes_same",
     "default_index",
+    "safe_sort_index",
 ]
 
 
@@ -157,16 +158,7 @@ def _get_combined_index(
         index = ensure_index(index)
 
     if sort:
-        try:
-            array_sorted = safe_sort(index)
-            array_sorted = cast(np.ndarray, array_sorted)
-            if isinstance(index, MultiIndex):
-                index = MultiIndex.from_tuples(array_sorted, names=index.names)
-            else:
-                index = Index(array_sorted, name=index.name)
-        except TypeError:
-            pass
-
+        index = safe_sort_index(index)
     # GH 29879
     if copy:
         index = index.copy()
@@ -174,6 +166,34 @@ def _get_combined_index(
     return index
 
 
+def safe_sort_index(index: Index) -> Index:
+    """
+    Returns the sorted index
+
+    We keep the dtypes and the name attributes.
+
+    Parameters
+    ----------
+    index : an Index
+
+    Returns
+    -------
+    Index
+    """
+    try:
+        array_sorted = safe_sort(index)
+    except TypeError:
+        pass
+    else:
+        array_sorted = cast(np.ndarray, array_sorted)
+        if isinstance(index, MultiIndex):
+            index = MultiIndex.from_tuples(array_sorted, names=index.names)
+        else:
+            index = Index(array_sorted, name=index.name, dtype=index.dtype)
+
+    return index
+
+
 def union_indexes(indexes, sort: bool | None = True) -> Index:
     """
     Return the union of indexes.
diff --git a/pandas/tests/util/test_assert_index_equal.py b/pandas/tests/util/test_assert_index_equal.py
index e3461e62b4eda..1fa7b979070a7 100644
--- a/pandas/tests/util/test_assert_index_equal.py
+++ b/pandas/tests/util/test_assert_index_equal.py
@@ -238,6 +238,14 @@ def test_index_equal_range_categories(check_categorical, exact):
         )
 
 
+def test_assert_index_equal_different_names_check_order_false():
+    # GH#47328
+    idx1 = Index([1, 3], name="a")
+    idx2 = Index([3, 1], name="b")
+    with pytest.raises(AssertionError, match='"names" are different'):
+        tm.assert_index_equal(idx1, idx2, check_order=False, check_names=True)
+
+
 def test_assert_index_equal_mixed_dtype():
     # GH#39168
     idx = Index(["foo", "bar", 42])