diff --git a/lib/matplotlib/tests/test_triangulation.py b/lib/matplotlib/tests/test_triangulation.py
index 74c79f33e22e..751b8c63cbae 100644
--- a/lib/matplotlib/tests/test_triangulation.py
+++ b/lib/matplotlib/tests/test_triangulation.py
@@ -985,6 +985,29 @@ def test_trirefiner_fortran_contiguous_triangles():
     assert_array_equal(fine_triang1.triangles, fine_triang2.triangles)
 
 
+def test_qhull_triangle_orientation():
+    # github issue 4437.
+    xi = np.linspace(-2, 2, 100)
+    x, y = map(np.ravel, np.meshgrid(xi, xi))
+    w = np.logical_and(x > y - 1, np.logical_and(x < -1.95, y > -1.2))
+    x, y = x[w], y[w]
+    theta = np.radians(25)
+    x1 = x*np.cos(theta) - y*np.sin(theta)
+    y1 = x*np.sin(theta) + y*np.cos(theta)
+
+    # Calculate Delaunay triangulation using Qhull.
+    triang = mtri.Triangulation(x1, y1)
+
+    # Neighbors returned by Qhull.
+    qhull_neighbors = triang.neighbors
+
+    # Obtain neighbors using own C++ calculation.
+    triang._neighbors = None
+    own_neighbors = triang.neighbors
+
+    assert_array_equal(qhull_neighbors, own_neighbors)
+
+
 if __name__ == '__main__':
     import nose
     nose.runmodule(argv=['-s', '--with-doctest'], exit=False)
diff --git a/lib/matplotlib/tri/_tri.cpp b/lib/matplotlib/tri/_tri.cpp
index d7d676032912..c51c1c1ced87 100644
--- a/lib/matplotlib/tri/_tri.cpp
+++ b/lib/matplotlib/tri/_tri.cpp
@@ -224,7 +224,8 @@ Triangulation::Triangulation(PyArrayObject* x,
                              PyArrayObject* triangles,
                              PyArrayObject* mask,
                              PyArrayObject* edges,
-                             PyArrayObject* neighbors)
+                             PyArrayObject* neighbors,
+                             int correct_triangle_orientations)
     : _npoints(PyArray_DIM(x,0)),
       _ntri(PyArray_DIM(triangles,0)),
       _x(x),
@@ -235,7 +236,8 @@ Triangulation::Triangulation(PyArrayObject* x,
       _neighbors(neighbors)
 {
     _VERBOSE("Triangulation::Triangulation");
-    correct_triangles();
+    if (correct_triangle_orientations)
+        correct_triangles();
 }
 
 Triangulation::~Triangulation()
@@ -2230,7 +2232,7 @@ TriModule::TriModule()
 Py::Object TriModule::new_triangulation(const Py::Tuple &args)
 {
     _VERBOSE("TriModule::new_triangulation");
-    args.verify_length(6);
+    args.verify_length(7);
 
     // x and y.
     PyArrayObject* x = (PyArrayObject*)PyArray_ContiguousFromObject(
@@ -2305,7 +2307,10 @@ Py::Object TriModule::new_triangulation(const Py::Tuple &args)
         }
     }
 
-    return Py::asObject(new Triangulation(x, y, triangles, mask, edges, neighbors));
+    int correct_triangle_orientations = Py::Int(args[6]);
+
+    return Py::asObject(new Triangulation(x, y, triangles, mask, edges, neighbors,
+                                          correct_triangle_orientations));
 }
 
 Py::Object TriModule::new_tricontourgenerator(const Py::Tuple &args)
diff --git a/lib/matplotlib/tri/_tri.h b/lib/matplotlib/tri/_tri.h
index c923411d9900..db75452c9881 100644
--- a/lib/matplotlib/tri/_tri.h
+++ b/lib/matplotlib/tri/_tri.h
@@ -180,13 +180,17 @@ class Triangulation : public Py::PythonExtension<Triangulation>
      *          once.
      *   neighbors: Optional int array of shape (ntri,3) indicating which
      *              triangles are the neighbors of which TriEdges, or -1 if
-     *              there is no such neighbor. */
+     *              there is no such neighbor.
+     *   correct_triangle_orientations: Whether or not should correct triangle
+     *                                  orientations so that vertices are
+     *                                  ordered anticlockwise. */
     Triangulation(PyArrayObject* x,
                   PyArrayObject* y,
                   PyArrayObject* triangles,
                   PyArrayObject* mask,
                   PyArrayObject* edges,
-                  PyArrayObject* neighbors);
+                  PyArrayObject* neighbors,
+                  int correct_triangle_orientations);
 
     virtual ~Triangulation();
 
diff --git a/lib/matplotlib/tri/triangulation.py b/lib/matplotlib/tri/triangulation.py
index 3309823e136a..69c4c153e715 100644
--- a/lib/matplotlib/tri/triangulation.py
+++ b/lib/matplotlib/tri/triangulation.py
@@ -107,7 +107,7 @@ def get_cpp_triangulation(self):
         if self._cpp_triangulation is None:
             self._cpp_triangulation = _tri.Triangulation(
                 self.x, self.y, self.triangles, self.mask, self._edges,
-                self._neighbors)
+                self._neighbors, not self.is_delaunay)
         return self._cpp_triangulation
 
     def get_masked_triangles(self):