I would never dare posting this here, but the question https://math.stackexchange.com/q/3019853/214353 on math.SE did not receive any feedback (except for 13 views and one upvote) since November 30, so...

I believe I've seen an answer to this somewhere (some Deligne notes?) but cannot recover it.

In the $\mathbb Z/2$-graded formulation, supercommutativity means that two odd degree elements anticommute while all other kinds of homogeneous elements commute.

If 2 is invertible, to name a $\mathbb Z/2$-grading is the same as to name an involution: for $\tau:A\to A$ with $\tau^2$ equal to identity, even elements are those with $\tau e=e$ and the odd ones those with $\tau o=-o$. Then every $a$ is uniquely the sum of the even element $(a+\tau a)/2$ and the odd element $(a-\tau a)/2$.

Now suppose 2 is *not* invertible, and we still want to formulate supercommutativity in terms of $\tau$ alone, in such a way that it becomes precisely supercommutativity as soon as 2 becomes invertible. How to do it?

My attempts produced ugly and inconclusive things like $$ 2yx=xy+x\tau(y)+\tau(x)y-\tau(x)\tau(y) $$ or $$ yx-\tau(y)x-y\tau(x)+\tau(y)\tau(x)=-xy+x\tau(y)+\tau(x)y-\tau(x)\tau(y). $$

There must be something much better, what is it?