  We present an approach to learning regular spatial transformations between
image pairs in the context of medical image registration. Contrary to
optimization-based registration techniques and many modern learning-based
methods, we do not directly penalize transformation irregularities but instead
promote transformation regularity via an inverse consistency penalty. We use a
neural network to predict a map between a source and a target image as well as
the map when swapping the source and target images. Different from existing
approaches, we compose these two resulting maps and regularize deviations of
the $\bf{Jacobian}$ of this composition from the identity matrix. This
regularizer -- $\texttt{GradICON}$ -- results in much better convergence when
training registration models compared to promoting inverse consistency of the
composition of maps directly while retaining the desirable implicit
regularization effects of the latter. We achieve state-of-the-art registration
performance on a variety of real-world medical image datasets using a single
set of hyperparameters and a single non-dataset-specific training protocol.
