Groups of rotating squares

This paper discusses the permutations that are generated by rotating $k \times k$ blocks of squares in a union of overlapping $k \times (k+1)$ rectangles. It is found that the single-rotation parity constraints effectively determine the group of accessible permutations. If there are $n$ squares, and the space is partitioned as a checkerboard with $m$ squares shaded and $n-m$ squares unshaded, then the four possible cases are $A_n$, $S_n$, $A_m \times A_{n-m}$, and the subgroup of all even permutations in $S_m \times S_{n-m}$, with exceptions when $k = 2$ and $k = 3$.