  We study the dynamics of the piecewise planar rotations
$F_{\lambda}(z)=\lambda (z-H(z)), $ with $z\in\C$, $H(z)=1$ if
$\mathrm{Im}(z)\ge0,$ $H(z)=-1$ if $\mathrm{Im}(z)<0,$ and
$\lambda=\mathrm{e}^{i \alpha} \in\C$, being $\alpha$ a rational multiple of
$\pi$. Our main results establish the dynamics in the so called regular set,
which is the complementary of the closure of the set formed by the preimages of
the discontinuity line. We prove that any connected component of this set is
open, bounded and periodic under the action of $F_\lambda$, with a period
$\ell,$ that depends on the connected component. Furthermore, $F_\lambda^\ell $
restricted to each component acts as a rotation with a period which also
depends on the connected component. As a consequence, any point in the regular
set is periodic. Among other results, we also prove that for any connected
component of the regular set, its boundary is a convex polygon with certain
maximum number of sides.
