diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index 28ff5a8bacc71..2dc13ff2fd682 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -686,8 +686,11 @@ def intersection(self, other, sort=False):
         """
         self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
+        other, _ = self._convert_can_do_setop(other)
 
         if self.equals(other):
+            if self.has_duplicates:
+                return self.unique()._get_reconciled_name_object(other)
             return self._get_reconciled_name_object(other)
 
         if len(self) == 0:
diff --git a/pandas/tests/indexes/datetimes/test_setops.py b/pandas/tests/indexes/datetimes/test_setops.py
index c8edd30e3f7aa..0127493888214 100644
--- a/pandas/tests/indexes/datetimes/test_setops.py
+++ b/pandas/tests/indexes/datetimes/test_setops.py
@@ -474,7 +474,7 @@ def test_intersection_list(self):
         values = [pd.Timestamp("2020-01-01"), pd.Timestamp("2020-02-01")]
         idx = DatetimeIndex(values, name="a")
         res = idx.intersection(values)
-        tm.assert_index_equal(res, idx.rename(None))
+        tm.assert_index_equal(res, idx)
 
     def test_month_range_union_tz_pytz(self, sort):
         from pytz import timezone
@@ -509,6 +509,20 @@ def test_month_range_union_tz_dateutil(self, sort):
 
         early_dr.union(late_dr, sort=sort)
 
+    @pytest.mark.parametrize("sort", [False, None])
+    def test_intersection_duplicates(self, sort):
+        # GH#38196
+        idx1 = Index(
+            [
+                pd.Timestamp("2019-12-13"),
+                pd.Timestamp("2019-12-12"),
+                pd.Timestamp("2019-12-12"),
+            ]
+        )
+        result = idx1.intersection(idx1, sort=sort)
+        expected = Index([pd.Timestamp("2019-12-13"), pd.Timestamp("2019-12-12")])
+        tm.assert_index_equal(result, expected)
+
 
 class TestCustomDatetimeIndex:
     def setup_method(self, method):