BUG: Index.is_monotonic... incorrectly caching Index.is_unique when first value is NaT

doc/source/whatsnew/v2.2.0.rst

@@ -323,6 +323,7 @@ Datetimelike
 ^^^^^^^^^^^^
 - Bug in :func:`concat` raising ``AttributeError`` when concatenating all-NA DataFrame with :class:`DatetimeTZDtype` dtype DataFrame. (:issue:`52093`)
 - Bug in :meth:`DatetimeIndex.union` returning object dtype for tz-aware indexes with the same timezone but different units (:issue:`55238`)
+- Bug in :meth:`Index.is_monotonic_increasing` and :meth:`Index.is_monotonic_decreasing` always caching :meth:`Index.is_unique` as ``True`` when first value in index is ``NaT`` (:issue:`55755`)
 - Bug in :meth:`Index.view` to a datetime64 dtype with non-supported resolution incorrectly raising (:issue:`55710`)
 - Bug in :meth:`Tick.delta` with very large ticks raising ``OverflowError`` instead of ``OutOfBoundsTimedelta`` (:issue:`55503`)
 - Bug in adding or subtracting a :class:`Week` offset to a ``datetime64`` :class:`Series`, :class:`Index`, or :class:`DataFrame` column with non-nanosecond resolution returning incorrect results (:issue:`55583`)

pandas/_libs/algos.pyx

@@ -814,7 +814,7 @@ def is_monotonic(ndarray[numeric_object_t, ndim=1] arr, bint timelike):
         return True, True, True
 
     if timelike and <int64_t>arr[0] == NPY_NAT:
-        return False, False, True
+        return False, False, False
 
     if numeric_object_t is not object:
         with nogil:

pandas/tests/indexes/test_base.py

@@ -1454,6 +1454,20 @@ def test_is_monotonic_na(self, index):
         assert index._is_strictly_monotonic_increasing is False
         assert index._is_strictly_monotonic_decreasing is False
 
+    @pytest.mark.parametrize("dtype", ["f8", "m8[ns]", "M8[us]"])
+    @pytest.mark.parametrize("unique_first", [True, False])
+    def test_is_monotonic_unique_na(self, dtype, unique_first):
+        # GH 55755
+        index = Index([None, 1, 1], dtype=dtype)
+        if unique_first:
+            assert index.is_unique is False
+            assert index.is_monotonic_increasing is False
+            assert index.is_monotonic_decreasing is False
+        else:
+            assert index.is_monotonic_increasing is False
+            assert index.is_monotonic_decreasing is False
+            assert index.is_unique is False
+
     def test_int_name_format(self, frame_or_series):
         index = Index(["a", "b", "c"], name=0)
         result = frame_or_series(list(range(3)), index=index)