Twisted aughts of alternating involutions

Let $\mathcal{M}(n)$ be the subgroup of $GL(n,\mathbb{Z})$ generated by the particular involutions that are identical to the identity, except for a single line where $-1$ and $+1$ alternate. We study the properties of $\mathcal{M}(n)$, and then find several notable characteristics of the unions of trajectories obtained by iteratively applying a fixed sequence of such involutions to elements from ${\mathbb{Z}}^n$.