  The noncommutative projective scheme $\operatorname{\mathsf{Proj_{nc}}} S$ of
a $(\pm 1)$-skew polynomial algebra $S$ in $n$ variables is considered to be a
$(\pm 1)$-skew projective space of dimension $n-1$. In this paper, using
combinatorial methods, we give a classification theorem for $(\pm 1)$-skew
projective spaces. Specifically, among other equivalences, we prove that $(\pm
1)$-skew projective spaces $\operatorname{\mathsf{Proj_{nc}}} S$ and
$\operatorname{\mathsf{Proj_{nc}}} S'$ are isomorphic if and only if certain
graphs associated to $S$ and $S'$ are switching (or mutation) equivalent. We
also discuss invariants of $(\pm 1)$-skew projective spaces from a
combinatorial point of view.
