  We investigate alternating sign matrices that are not permutation matrices,
but have finite order in a general linear group. We classify all such examples
of the form $P+T$, where $P$ is a permutation matrix and $T$ has four non-zero
entries, forming a square with entries $1$ and $-1$ in each row and column. We
show that the multiplicative orders of these matrices do not always coincide
with those of permutation matrices of the same size. We pose the problem of
identifying finite subgroups of general linear groups that are generated by
alternating sign matrices.
