diff --git a/lib/matplotlib/colorbar.py b/lib/matplotlib/colorbar.py
index b1cd08d955a7..16b0377a43bc 100644
--- a/lib/matplotlib/colorbar.py
+++ b/lib/matplotlib/colorbar.py
@@ -1006,24 +1006,34 @@ def _reset_locator_formatter_scale(self):
         self.locator = None
         self.minorlocator = None
         self.formatter = None
-        if ((self.spacing == 'uniform') and
-            ((self.boundaries is not None) or
-              isinstance(self.norm, colors.BoundaryNorm))):
-            funcs = (self._forward_boundaries, self._inverse_boundaries)
-            self.ax.set_xscale('function', functions=funcs)
-            self.ax.set_yscale('function', functions=funcs)
-            self.__scale = 'function'
-        elif hasattr(self.norm, '_scale') and (self.norm._scale is not None):
+        if (self.boundaries is not None or
+                isinstance(self.norm, colors.BoundaryNorm)):
+            if self.spacing == 'uniform':
+                funcs = (self._forward_boundaries, self._inverse_boundaries)
+                self.ax.set_xscale('function', functions=funcs)
+                self.ax.set_yscale('function', functions=funcs)
+                self.__scale = 'function'
+            elif self.spacing == 'proportional':
+                self.__scale = 'linear'
+                self.ax.set_xscale('linear')
+                self.ax.set_yscale('linear')
+        elif hasattr(self.norm, '_scale') and self.norm._scale is not None:
+            # use the norm's scale:
             self.ax.set_xscale(self.norm._scale)
             self.ax.set_yscale(self.norm._scale)
             self.__scale = self.norm._scale.name
-        else:
+        elif type(self.norm) is colors.Normalize:
+            # plain Normalize:
             self.ax.set_xscale('linear')
             self.ax.set_yscale('linear')
-            if type(self.norm) is colors.Normalize:
-                self.__scale = 'linear'
-            else:
-                self.__scale = 'manual'
+            self.__scale = 'linear'
+        else:
+            # norm._scale is None or not an attr: derive the scale from
+            # the Norm:
+            funcs = (self.norm, self.norm.inverse)
+            self.ax.set_xscale('function', functions=funcs)
+            self.ax.set_yscale('function', functions=funcs)
+            self.__scale = 'function'
 
     def _locate(self, x):
         """
diff --git a/lib/matplotlib/colors.py b/lib/matplotlib/colors.py
index f678a4ffefd5..3b7e5988eab7 100644
--- a/lib/matplotlib/colors.py
+++ b/lib/matplotlib/colors.py
@@ -1152,7 +1152,7 @@ def __init__(self, vmin=None, vmax=None, clip=False):
         self.vmin = _sanitize_extrema(vmin)
         self.vmax = _sanitize_extrema(vmax)
         self.clip = clip
-        self._scale = scale.LinearScale(axis=None)
+        self._scale = None  # will default to LinearScale for colorbar
 
     @staticmethod
     def process_value(value):
@@ -1334,6 +1334,16 @@ def __call__(self, value, clip=None):
             result = np.atleast_1d(result)[0]
         return result
 
+    def inverse(self, value):
+        if not self.scaled():
+            raise ValueError("Not invertible until both vmin and vmax are set")
+        (vmin,), _ = self.process_value(self.vmin)
+        (vmax,), _ = self.process_value(self.vmax)
+        (vcenter,), _ = self.process_value(self.vcenter)
+
+        result = np.interp(value, [0, 0.5, 1.], [vmin, vcenter, vmax])
+        return result
+
 
 class CenteredNorm(Normalize):
     def __init__(self, vcenter=0, halfrange=None, clip=False):
diff --git a/lib/matplotlib/tests/baseline_images/test_colorbar/colorbar_twoslope.png b/lib/matplotlib/tests/baseline_images/test_colorbar/colorbar_twoslope.png
new file mode 100644
index 000000000000..27d9227dcfe0
Binary files /dev/null and b/lib/matplotlib/tests/baseline_images/test_colorbar/colorbar_twoslope.png differ
diff --git a/lib/matplotlib/tests/test_colorbar.py b/lib/matplotlib/tests/test_colorbar.py
index ac054755c4c1..6167618575dd 100644
--- a/lib/matplotlib/tests/test_colorbar.py
+++ b/lib/matplotlib/tests/test_colorbar.py
@@ -771,3 +771,17 @@ def test_inset_colorbar_layout():
     np.testing.assert_allclose(cb.ax.get_position().bounds,
                                [0.87, 0.342, 0.0237, 0.315], atol=0.01)
     assert cb.ax.outer_ax in ax.child_axes
+
+
+@image_comparison(['colorbar_twoslope.png'], remove_text=True,
+                  style='mpl20')
+def test_twoslope_colorbar():
+    # Note that the first tick = 20, and should be in the middle
+    # of the colorbar (white)
+    fig, ax = plt.subplots()
+
+    norm = mcolors.TwoSlopeNorm(20, 0, 100)
+    pc = ax.pcolormesh(np.arange(1, 11), np.arange(1, 11),
+                       np.arange(100).reshape(10, 10),
+                       norm=norm, cmap='RdBu_r')
+    fig.colorbar(pc)
diff --git a/lib/matplotlib/tests/test_colors.py b/lib/matplotlib/tests/test_colors.py
index 81dd65bab713..ae004e957591 100644
--- a/lib/matplotlib/tests/test_colors.py
+++ b/lib/matplotlib/tests/test_colors.py
@@ -1406,6 +1406,5 @@ def test_norm_deepcopy():
     norm = mcolors.Normalize()
     norm.vmin = 0.0002
     norm2 = copy.deepcopy(norm)
-    assert isinstance(norm2._scale, mscale.LinearScale)
+    assert norm2._scale is None
     assert norm2.vmin == norm.vmin
-    assert norm2._scale is not norm._scale