From 5ae4d915d8a4e002585f29dc562d49c3ceabfbf6 Mon Sep 17 00:00:00 2001
From: Patrick Hoefler <patrick_hoefler@gmx.net>
Date: Tue, 28 Feb 2023 01:48:38 +0100
Subject: [PATCH 1/2] BUG: concat sorting when sort=False for all dti

---
 doc/source/whatsnew/v2.1.0.rst                |  2 +-
 pandas/core/indexes/api.py                    |  2 +-
 pandas/tests/reshape/concat/test_datetimes.py | 12 ++++++------
 3 files changed, 8 insertions(+), 8 deletions(-)

diff --git a/doc/source/whatsnew/v2.1.0.rst b/doc/source/whatsnew/v2.1.0.rst
index 45b5c16415f9d..311712db5bd7b 100644
--- a/doc/source/whatsnew/v2.1.0.rst
+++ b/doc/source/whatsnew/v2.1.0.rst
@@ -191,7 +191,7 @@ Groupby/resample/rolling
 
 Reshaping
 ^^^^^^^^^
--
+- Bug in :func:`concat` sorting :class:`Index` when only :class:`DatetimeIndex` is given (:issue:`51210`)
 -
 
 Sparse
diff --git a/pandas/core/indexes/api.py b/pandas/core/indexes/api.py
index fcf529f5be9ac..6b68cc1a5089f 100644
--- a/pandas/core/indexes/api.py
+++ b/pandas/core/indexes/api.py
@@ -276,7 +276,7 @@ def _find_common_index_dtype(inds):
             raise TypeError("Cannot join tz-naive with tz-aware DatetimeIndex")
 
         if len(dtis) == len(indexes):
-            sort = True
+            sort = sort
             result = indexes[0]
 
         elif len(dtis) > 1:
diff --git a/pandas/tests/reshape/concat/test_datetimes.py b/pandas/tests/reshape/concat/test_datetimes.py
index f16358813488e..c888a8ddf34e2 100644
--- a/pandas/tests/reshape/concat/test_datetimes.py
+++ b/pandas/tests/reshape/concat/test_datetimes.py
@@ -77,23 +77,23 @@ def test_concat_datetime_timezone(self):
 
         exp_idx = DatetimeIndex(
             [
-                "2010-12-31 15:00:00+00:00",
-                "2010-12-31 16:00:00+00:00",
-                "2010-12-31 17:00:00+00:00",
                 "2010-12-31 23:00:00+00:00",
                 "2011-01-01 00:00:00+00:00",
                 "2011-01-01 01:00:00+00:00",
+                "2010-12-31 15:00:00+00:00",
+                "2010-12-31 16:00:00+00:00",
+                "2010-12-31 17:00:00+00:00",
             ]
         )
 
         expected = DataFrame(
             [
-                [np.nan, 1],
-                [np.nan, 2],
-                [np.nan, 3],
                 [1, np.nan],
                 [2, np.nan],
                 [3, np.nan],
+                [np.nan, 1],
+                [np.nan, 2],
+                [np.nan, 3],
             ],
             index=exp_idx,
             columns=["a", "b"],

From cf3b2b780423f33c45ba667ab42a925e4ea0dc84 Mon Sep 17 00:00:00 2001
From: Patrick Hoefler <patrick_hoefler@gmx.net>
Date: Tue, 28 Feb 2023 15:56:51 +0100
Subject: [PATCH 2/2] Fix pylint

---
 pandas/core/indexes/api.py | 1 -
 1 file changed, 1 deletion(-)

diff --git a/pandas/core/indexes/api.py b/pandas/core/indexes/api.py
index 6b68cc1a5089f..68df699d58a45 100644
--- a/pandas/core/indexes/api.py
+++ b/pandas/core/indexes/api.py
@@ -276,7 +276,6 @@ def _find_common_index_dtype(inds):
             raise TypeError("Cannot join tz-naive with tz-aware DatetimeIndex")
 
         if len(dtis) == len(indexes):
-            sort = sort
             result = indexes[0]
 
         elif len(dtis) > 1: