diff --git a/doc/whats-new.rst b/doc/whats-new.rst
index c8ae5ac43c8..de7e6c8f6ff 100644
--- a/doc/whats-new.rst
+++ b/doc/whats-new.rst
@@ -71,7 +71,7 @@ Enhancements
   :py:meth:`~xarray.Dataset.differentiate`,
   :py:meth:`~xarray.DataArray.interp`, and
   :py:meth:`~xarray.Dataset.interp`.
-  By `Spencer Clark <https://github.com/spencerkclark>`_
+  By `Spencer Clark <https://github.com/spencerkclark>`_.
   
 Bug fixes
 ~~~~~~~~~
@@ -97,6 +97,13 @@ Bug fixes
   ``open_rasterio`` to error (:issue:`2454`).
   By `Stephan Hoyer <https://github.com/shoyer>`_.
 
+- Subtracting one CFTimeIndex from another now returns a
+  ``pandas.TimedeltaIndex``, analogous to the behavior for DatetimeIndexes
+  (:issue:`2484`).  By `Spencer Clark <https://github.com/spencerkclark>`_.
+- Adding a TimedeltaIndex to, or subtracting a TimedeltaIndex from a
+  CFTimeIndex is now allowed (:issue:`2484`).
+  By `Spencer Clark <https://github.com/spencerkclark>`_. 
+
 .. _whats-new.0.10.9:
 
 v0.10.9 (21 September 2018)
diff --git a/xarray/coding/cftimeindex.py b/xarray/coding/cftimeindex.py
index dea896c199a..5de055c1b9a 100644
--- a/xarray/coding/cftimeindex.py
+++ b/xarray/coding/cftimeindex.py
@@ -359,13 +359,27 @@ def shift(self, n, freq):
                 "str or datetime.timedelta, got {}.".format(freq))
 
     def __add__(self, other):
+        if isinstance(other, pd.TimedeltaIndex):
+            other = other.to_pytimedelta()
         return CFTimeIndex(np.array(self) + other)
 
     def __radd__(self, other):
+        if isinstance(other, pd.TimedeltaIndex):
+            other = other.to_pytimedelta()
         return CFTimeIndex(other + np.array(self))
 
     def __sub__(self, other):
-        return CFTimeIndex(np.array(self) - other)
+        if isinstance(other, CFTimeIndex):
+            return pd.TimedeltaIndex(np.array(self) - np.array(other))
+        elif isinstance(other, pd.TimedeltaIndex):
+            return CFTimeIndex(np.array(self) - other.to_pytimedelta())
+        else:
+            return CFTimeIndex(np.array(self) - other)
+
+    def _add_delta(self, deltas):
+        # To support TimedeltaIndex + CFTimeIndex with older versions of
+        # pandas.  No longer used as of pandas 0.23.
+        return self + deltas
 
 
 def _parse_iso8601_without_reso(date_type, datetime_str):
diff --git a/xarray/tests/test_cftimeindex.py b/xarray/tests/test_cftimeindex.py
index d1726ab3313..e18c55d2fae 100644
--- a/xarray/tests/test_cftimeindex.py
+++ b/xarray/tests/test_cftimeindex.py
@@ -629,6 +629,17 @@ def test_cftimeindex_add(index):
     assert isinstance(result, CFTimeIndex)
 
 
+@pytest.mark.skipif(not has_cftime, reason='cftime not installed')
+@pytest.mark.parametrize('calendar', _CFTIME_CALENDARS)
+def test_cftimeindex_add_timedeltaindex(calendar):
+    a = xr.cftime_range('2000', periods=5, calendar=calendar)
+    deltas = pd.TimedeltaIndex([timedelta(days=2) for _ in range(5)])
+    result = a + deltas
+    expected = a.shift(2, 'D')
+    assert result.equals(expected)
+    assert isinstance(result, CFTimeIndex)
+
+
 @pytest.mark.skipif(not has_cftime, reason='cftime not installed')
 def test_cftimeindex_radd(index):
     date_type = index.date_type
@@ -640,6 +651,17 @@ def test_cftimeindex_radd(index):
     assert isinstance(result, CFTimeIndex)
 
 
+@pytest.mark.skipif(not has_cftime, reason='cftime not installed')
+@pytest.mark.parametrize('calendar', _CFTIME_CALENDARS)
+def test_timedeltaindex_add_cftimeindex(calendar):
+    a = xr.cftime_range('2000', periods=5, calendar=calendar)
+    deltas = pd.TimedeltaIndex([timedelta(days=2) for _ in range(5)])
+    result = deltas + a
+    expected = a.shift(2, 'D')
+    assert result.equals(expected)
+    assert isinstance(result, CFTimeIndex)
+
+
 @pytest.mark.skipif(not has_cftime, reason='cftime not installed')
 def test_cftimeindex_sub(index):
     date_type = index.date_type
@@ -652,6 +674,28 @@ def test_cftimeindex_sub(index):
     assert isinstance(result, CFTimeIndex)
 
 
+@pytest.mark.skipif(not has_cftime, reason='cftime not installed')
+@pytest.mark.parametrize('calendar', _CFTIME_CALENDARS)
+def test_cftimeindex_sub_cftimeindex(calendar):
+    a = xr.cftime_range('2000', periods=5, calendar=calendar)
+    b = a.shift(2, 'D')
+    result = b - a
+    expected = pd.TimedeltaIndex([timedelta(days=2) for _ in range(5)])
+    assert result.equals(expected)
+    assert isinstance(result, pd.TimedeltaIndex)
+
+
+@pytest.mark.skipif(not has_cftime, reason='cftime not installed')
+@pytest.mark.parametrize('calendar', _CFTIME_CALENDARS)
+def test_cftimeindex_sub_timedeltaindex(calendar):
+    a = xr.cftime_range('2000', periods=5, calendar=calendar)
+    deltas = pd.TimedeltaIndex([timedelta(days=2) for _ in range(5)])
+    result = a - deltas
+    expected = a.shift(-2, 'D')
+    assert result.equals(expected)
+    assert isinstance(result, CFTimeIndex)
+
+
 @pytest.mark.skipif(not has_cftime, reason='cftime not installed')
 def test_cftimeindex_rsub(index):
     with pytest.raises(TypeError):