Consider the (single-sorted) algebraic theory with one binary operation symbol $\ast$ and the following single axiom: $$(x_1 \ast x_2) \ast (x_3 \ast x) = x.$$ One possible model for this theory comes from $\mathbb{Z}$, where $m \ast n := -n$ for all $m, n \in \mathbb{Z}$ (i.e. $\ast$ is interpreted as the twisted projection on $\mathbb{Z}$).
Has this theory been studied before? And more specifically, is anything known about its word problem?