diff --git a/doc/source/whatsnew/v0.23.0.txt b/doc/source/whatsnew/v0.23.0.txt
index 14949267fc37d..9df7563d3b869 100644
--- a/doc/source/whatsnew/v0.23.0.txt
+++ b/doc/source/whatsnew/v0.23.0.txt
@@ -273,6 +273,7 @@ Other API Changes
 - The default ``Timedelta`` constructor now accepts an ``ISO 8601 Duration`` string as an argument (:issue:`19040`)
 - ``IntervalDtype`` now returns ``True`` when compared against ``'interval'`` regardless of subtype, and ``IntervalDtype.name`` now returns ``'interval'`` regardless of subtype (:issue:`18980`)
 - :func:`Series.to_csv` now accepts a ``compression`` argument that works in the same way as the ``compression`` argument in :func:`DataFrame.to_csv` (:issue:`18958`)
+- Addition or subtraction of ``NaT`` from :class:`TimedeltaIndex` will return ``TimedeltaIndex`` instead of ``DatetimeIndex`` (:issue:`19124`)
 
 .. _whatsnew_0230.deprecations:
 
diff --git a/pandas/core/indexes/timedeltas.py b/pandas/core/indexes/timedeltas.py
index 12ca26cfe0266..866329b16c830 100644
--- a/pandas/core/indexes/timedeltas.py
+++ b/pandas/core/indexes/timedeltas.py
@@ -403,7 +403,8 @@ def _add_datelike(self, other):
         # adding a timedeltaindex to a datetimelike
         from pandas import Timestamp, DatetimeIndex
         if other is NaT:
-            result = self._nat_new(box=False)
+            # GH#19124 pd.NaT is treated like a timedelta
+            return self._nat_new()
         else:
             other = Timestamp(other)
             i8 = self.asi8
@@ -413,12 +414,13 @@ def _add_datelike(self, other):
         return DatetimeIndex(result, name=self.name, copy=False)
 
     def _sub_datelike(self, other):
-        from pandas import DatetimeIndex
+        # GH#19124 Timedelta - datetime is not in general well-defined.
+        # We make an exception for pd.NaT, which in this case quacks
+        # like a timedelta.
         if other is NaT:
-            result = self._nat_new(box=False)
+            return self._nat_new()
         else:
             raise TypeError("cannot subtract a datelike from a TimedeltaIndex")
-        return DatetimeIndex(result, name=self.name, copy=False)
 
     def _add_offset_array(self, other):
         # Array/Index of DateOffset objects
diff --git a/pandas/tests/scalar/test_nat.py b/pandas/tests/scalar/test_nat.py
index d0d204253e3f1..73e8c783ba882 100644
--- a/pandas/tests/scalar/test_nat.py
+++ b/pandas/tests/scalar/test_nat.py
@@ -302,14 +302,30 @@ def test_nat_arithmetic_index():
         tm.assert_index_equal(left - right, exp)
         tm.assert_index_equal(right - left, exp)
 
-    # timedelta
+    # timedelta # GH#19124
     tdi = TimedeltaIndex(['1 day', '2 day'], name='x')
-    exp = DatetimeIndex([NaT, NaT], name='x')
-    for (left, right) in [(NaT, tdi)]:
-        tm.assert_index_equal(left + right, exp)
-        tm.assert_index_equal(right + left, exp)
-        tm.assert_index_equal(left - right, exp)
-        tm.assert_index_equal(right - left, exp)
+    tdi_nat = TimedeltaIndex([NaT, NaT], name='x')
+
+    tm.assert_index_equal(tdi + NaT, tdi_nat)
+    tm.assert_index_equal(NaT + tdi, tdi_nat)
+    tm.assert_index_equal(tdi - NaT, tdi_nat)
+    tm.assert_index_equal(NaT - tdi, tdi_nat)
+
+
+@pytest.mark.parametrize('box, assert_func', [
+    (TimedeltaIndex, tm.assert_index_equal),
+    pytest.param(Series, tm.assert_series_equal,
+                 marks=pytest.mark.xfail(reason='NaT - Series returns NaT'))
+])
+def test_nat_arithmetic_td64_vector(box, assert_func):
+    # GH#19124
+    vec = box(['1 day', '2 day'], dtype='timedelta64[ns]')
+    box_nat = box([NaT, NaT], dtype='timedelta64[ns]')
+
+    assert_func(vec + NaT, box_nat)
+    assert_func(NaT + vec, box_nat)
+    assert_func(vec - NaT, box_nat)
+    assert_func(NaT - vec, box_nat)
 
 
 def test_nat_pinned_docstrings():