diff --git a/nitransforms/base.py b/nitransforms/base.py
index 9a1600a0..87bda079 100644
--- a/nitransforms/base.py
+++ b/nitransforms/base.py
@@ -303,6 +303,10 @@ def apply(
 
         return resampled
 
+    def __matmul__(self, b):
+        """Compose with a transform on the right."""
+        return b
+
     def map(self, x, inverse=False):
         r"""
         Apply :math:`y = f(x)`.
diff --git a/nitransforms/nonlinear.py b/nitransforms/nonlinear.py
index 0c9bb371..b2bd5ad8 100644
--- a/nitransforms/nonlinear.py
+++ b/nitransforms/nonlinear.py
@@ -20,11 +20,12 @@
     ImageGrid,
     SpatialReference,
     _as_homogeneous,
+    EQUALITY_TOL,
 )
 
 
-class DisplacementsFieldTransform(TransformBase):
-    """Represents a dense field of displacements (one vector per voxel)."""
+class DeformationFieldTransform(TransformBase):
+    """Represents a dense field of deformed locations (corresponding to each voxel)."""
 
     __slots__ = ["_field"]
 
@@ -34,8 +35,8 @@ def __init__(self, field, reference=None):
 
         Example
         -------
-        >>> DisplacementsFieldTransform(test_dir / "someones_displacement_field.nii.gz")
-        <DisplacementFieldTransform[3D] (57, 67, 56)>
+        >>> DeformationFieldTransform(test_dir / "someones_displacement_field.nii.gz")
+        <DeformationFieldTransform[3D] (57, 67, 56)>
 
         """
         super().__init__()
@@ -59,13 +60,13 @@ def __init__(self, field, reference=None):
         ndim = self._field.ndim - 1
         if self._field.shape[-1] != ndim:
             raise TransformError(
-                "The number of components of the displacements (%d) does not "
+                "The number of components of the displacements (%d) does not match "
                 "the number of dimensions (%d)" % (self._field.shape[-1], ndim)
             )
 
     def __repr__(self):
         """Beautify the python representation."""
-        return f"<DisplacementFieldTransform[{self._field.shape[-1]}D] {self._field.shape[:3]}>"
+        return f"<{self.__class__.__name__}[{self._field.shape[-1]}D] {self._field.shape[:3]}>"
 
     def map(self, x, inverse=False):
         r"""
@@ -92,12 +93,12 @@ def map(self, x, inverse=False):
 
         Examples
         --------
-        >>> xfm = DisplacementsFieldTransform(test_dir / "someones_displacement_field.nii.gz")
+        >>> xfm = DeformationFieldTransform(test_dir / "someones_displacement_field.nii.gz")
         >>> xfm.map([-6.5, -36., -19.5]).tolist()
-        [[-6.5, -36.475167989730835, -19.5]]
+        [[0.0, -0.47516798973083496, 0.0]]
 
         >>> xfm.map([[-6.5, -36., -19.5], [-1., -41.5, -11.25]]).tolist()
-        [[-6.5, -36.475167989730835, -19.5], [-1.0, -42.038356602191925, -11.25]]
+        [[0.0, -0.47516798973083496, 0.0], [0.0, -0.538356602191925, 0.0]]
 
         """
 
@@ -108,7 +109,76 @@ def map(self, x, inverse=False):
         if np.any(np.abs(ijk - indexes) > 0.05):
             warnings.warn("Some coordinates are off-grid of the displacements field.")
         indexes = tuple(tuple(i) for i in indexes.T)
-        return x + self._field[indexes]
+        return self._field[indexes]
+
+    def __matmul__(self, b):
+        """
+        Compose with a transform on the right.
+
+        Examples
+        --------
+        >>> xfm = DeformationFieldTransform(test_dir / "someones_displacement_field.nii.gz")
+        >>> xfm2 = xfm @ TransformBase()
+        >>> xfm == xfm2
+        True
+
+        """
+        retval = b.map(
+            self._field.reshape((-1, self._field.shape[-1]))
+        ).reshape(self._field.shape)
+        return DeformationFieldTransform(retval, reference=self.reference)
+
+    def __eq__(self, other):
+        """
+        Overload equals operator.
+
+        Examples
+        --------
+        >>> xfm1 = DeformationFieldTransform(test_dir / "someones_displacement_field.nii.gz")
+        >>> xfm2 = DeformationFieldTransform(test_dir / "someones_displacement_field.nii.gz")
+        >>> xfm1 == xfm2
+        True
+
+        """
+        _eq = np.allclose(self._field, other._field, rtol=EQUALITY_TOL)
+        if _eq and self._reference != other._reference:
+            warnings.warn("Fields are equal, but references do not match.")
+        return _eq
+
+
+class DisplacementsFieldTransform(DeformationFieldTransform):
+    """
+    Represents a dense field of displacements (one vector per voxel).
+
+    Converting to a field of deformations is straightforward by just adding the corresponding
+    displacement to the :math:`(x, y, z)` coordinates of each voxel.
+    Numerically, deformation fields are less susceptible to rounding errors
+    than displacements fields.
+    SPM generally prefers deformations for that reason.
+
+    """
+
+    __slots__ = ["_displacements"]
+
+    def __init__(self, field, reference=None):
+        """
+        Create a transform supported by a field of voxel-wise displacements.
+
+        Example
+        -------
+        >>> xfm = DisplacementsFieldTransform(test_dir / "someones_displacement_field.nii.gz")
+        >>> xfm
+        <DisplacementsFieldTransform[3D] (57, 67, 56)>
+
+        >>> xfm.map([[-6.5, -36., -19.5], [-1., -41.5, -11.25]]).tolist()
+        [[-6.5, -36.47516632080078, -19.5], [-1.0, -42.03835678100586, -11.25]]
+
+        """
+        super().__init__(field, reference=reference)
+        self._displacements = self._field
+        # Convert from displacements to deformations fields
+        # (just add the origin to the displacements vector)
+        self._field += self.reference.ndcoords.T.reshape(self._field.shape)
 
     @classmethod
     def from_filename(cls, filename, fmt="X5"):