  In this paper we introduce some infinite rectangle exchange transformations
which are based on the simultaneous turning of the squares within a sequence of
square grids. We will show that such noncompact systems have higher dimensional
dynamical compactifications. In good cases, these compactifications are
polytope exchange transformations based on pairs of Euclidean lattices. In each
dimension $8m+4$ there is a $4m+2$ dimensional family of them. Here
$m=0,1,2,...$ The case $m=0$, which we studied in depth in an earlier paper,
has close connections to the $E_4$ Weyl group and the $(2,4,\infty)$ hyperbolic
triangle group.
