I have stumbled upon a construction that has probably been noticed before, and I wonder if anyone can point me to a reference.

Suppose that $K$ is a simplicial complex. Let $P(K)$ be the free abelian group on the set of simplices of $K$, all dimensions together. View it as a Grothendieck group of subcomplexes, so that for example the boundary $\partial \sigma$ of a simplex $\sigma$ appears in it as an alternating sum of the proper faces of $\sigma$. The open simplices $int (\sigma) =\sigma-\partial \sigma$ form another basis. Do not think about orientations.

Let $I$ be the involution of $P(K)$ given by $\sigma\mapsto (-1)^m\ int\ (\sigma)$ where $m$ is the dimension of $\sigma$.

It appears that the homology of the $2$-periodic chain complex given by $1-I:P(K)\to P(K)$ and $1+I:P(K)\to P(K)$ is isomorphic to the mod $2$ homology of $K$ made $2$-periodic.

EDIT: In case the slightly vague reference to a "Grothendieck group" is confusing, let me be more explicit: $P(K)$ has a basis consisting of the simplices of the complex. If $\sigma$ is a $p$-simplex of $K$, then the map $I$ takes the basis element $\sigma$ to the sum of $2^{p+1}-1$ terms $(-1)^{dim\ \tau}\tau$, one term for each simplex $\tau$ that is contained in $\sigma$, including $\sigma$ itself. Note that I never mentioned an orientation or an ordering of vertices.