  We extend the notion of Rauzy induction of interval exchange transformations
to the case of toral $\mathbb{Z}^2$-rotation, i.e., $\mathbb{Z}^2$-action
defined by rotations on a 2-torus. If $\mathcal{X}_{\mathcal{P},R}$ denotes the
symbolic dynamical system corresponding to a partition $\mathcal{P}$ and
$\mathbb{Z}^2$-action $R$ such that $R$ is Cartesian on a sub-domain $W$, we
express the 2-dimensional configurations in $\mathcal{X}_{\mathcal{P},R}$ as
the image under a $2$-dimensional morphism (up to a shift) of a configuration
in $\mathcal{X}_{\widehat{\mathcal{P}}|_W,\widehat{R}|_W}$ where
$\widehat{\mathcal{P}}|_W$ is the induced partition and $\widehat{R}|_W$ is the
induced $\mathbb{Z}^2$-action on $W$.
  We focus on one example $\mathcal{X}_{\mathcal{P}_0,R_0}$ for which we obtain
an eventually periodic sequence of 2-dimensional morphisms. We prove that it is
the same as the substitutive structure of the minimal subshift $X_0$ of the
Jeandel-Rao Wang shift computed in an earlier work by the author. As a
consequence, $\mathcal{P}_0$ is a Markov partition for the associated toral
$\mathbb{Z}^2$-rotation $R_0$. It also implies that the subshift $X_0$ is
uniquely ergodic and is isomorphic to the toral $\mathbb{Z}^2$-rotation $R_0$
which can be seen as a generalization for 2-dimensional subshifts of the
relation between Sturmian sequences and irrational rotations on a circle.
Batteries included: the algorithms and code to reproduce the proofs are
provided.
