diff --git a/doc/source/whatsnew/v0.23.0.txt b/doc/source/whatsnew/v0.23.0.txt
index 07e633ded942a..d7a3f0d077302 100644
--- a/doc/source/whatsnew/v0.23.0.txt
+++ b/doc/source/whatsnew/v0.23.0.txt
@@ -376,7 +376,8 @@ Conversion
 - Bug in :class:`Series` with ``dtype='timedelta64[ns]`` where addition or subtraction of ``TimedeltaIndex`` could return a ``Series`` with an incorrect name (issue:`19043`)
 - Fixed bug where comparing :class:`DatetimeIndex` failed to raise ``TypeError`` when attempting to compare timezone-aware and timezone-naive datetimelike objects (:issue:`18162`)
 - Bug in :class:`DatetimeIndex` where the repr was not showing high-precision time values at the end of a day (e.g., 23:59:59.999999999) (:issue:`19030`)
-
+- Bug where dividing a scalar timedelta-like object with :class:`TimedeltaIndex` performed the reciprocal operation (:issue:`19125`)
+-
 
 Indexing
 ^^^^^^^^
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 55a26d57fa1d6..f634d809560ee 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -3861,7 +3861,7 @@ def dropna(self, how='any'):
             return self._shallow_copy(self.values[~self._isnan])
         return self._shallow_copy()
 
-    def _evaluate_with_timedelta_like(self, other, op, opstr):
+    def _evaluate_with_timedelta_like(self, other, op, opstr, reversed=False):
         raise TypeError("can only perform ops with timedelta like values")
 
     def _evaluate_with_datetime_like(self, other, op, opstr):
@@ -4025,7 +4025,8 @@ def _evaluate_numeric_binop(self, other):
                 # handle time-based others
                 if isinstance(other, (ABCDateOffset, np.timedelta64,
                                       Timedelta, datetime.timedelta)):
-                    return self._evaluate_with_timedelta_like(other, op, opstr)
+                    return self._evaluate_with_timedelta_like(other, op, opstr,
+                                                              reversed)
                 elif isinstance(other, (Timestamp, np.datetime64)):
                     return self._evaluate_with_datetime_like(other, op, opstr)
 
diff --git a/pandas/core/indexes/timedeltas.py b/pandas/core/indexes/timedeltas.py
index ccedaa9cf71ee..12ca26cfe0266 100644
--- a/pandas/core/indexes/timedeltas.py
+++ b/pandas/core/indexes/timedeltas.py
@@ -372,7 +372,7 @@ def _add_delta(self, delta):
         result = TimedeltaIndex(new_values, freq='infer', name=name)
         return result
 
-    def _evaluate_with_timedelta_like(self, other, op, opstr):
+    def _evaluate_with_timedelta_like(self, other, op, opstr, reversed=False):
         if isinstance(other, ABCSeries):
             # GH#19042
             return NotImplemented
@@ -386,10 +386,14 @@ def _evaluate_with_timedelta_like(self, other, op, opstr):
                         "division by pd.NaT not implemented")
 
                 i8 = self.asi8
+                left, right = i8, other.value
+                if reversed:
+                    left, right = right, left
+
                 if opstr in ['__floordiv__']:
-                    result = i8 // other.value
+                    result = left // right
                 else:
-                    result = op(i8, float(other.value))
+                    result = op(left, float(right))
                 result = self._maybe_mask_results(result, convert='float64')
                 return Index(result, name=self.name, copy=False)
 
@@ -972,6 +976,7 @@ def _is_convertible_to_index(other):
 
 
 def _is_convertible_to_td(key):
+    # TODO: Not all DateOffset objects are convertible to Timedelta
     return isinstance(key, (DateOffset, timedelta, Timedelta,
                             np.timedelta64, compat.string_types))
 
diff --git a/pandas/tests/indexes/timedeltas/test_arithmetic.py b/pandas/tests/indexes/timedeltas/test_arithmetic.py
index 2581a8fad078a..3ec918e391860 100644
--- a/pandas/tests/indexes/timedeltas/test_arithmetic.py
+++ b/pandas/tests/indexes/timedeltas/test_arithmetic.py
@@ -286,6 +286,23 @@ def test_tdi_radd_timestamp(self):
 
     # -------------------------------------------------------------
 
+    @pytest.mark.parametrize('scalar_td', [
+        timedelta(minutes=10, seconds=7),
+        Timedelta('10m7s'),
+        Timedelta('10m7s').to_timedelta64()])
+    def test_tdi_floordiv_timedelta_scalar(self, scalar_td):
+        # GH#19125
+        tdi = TimedeltaIndex(['00:05:03', '00:05:03', pd.NaT], freq=None)
+        expected = pd.Index([2.0, 2.0, np.nan])
+
+        res = tdi.__rfloordiv__(scalar_td)
+        tm.assert_index_equal(res, expected)
+
+        expected = pd.Index([0.0, 0.0, np.nan])
+
+        res = tdi // (scalar_td)
+        tm.assert_index_equal(res, expected)
+
     # TODO: Split by operation, better name
     def test_ops_compat(self):