I don't know if this actually counts (since I don't know if this functional equation is actually useful...), but in the paper

R.M. Dicker, A set of independent axioms for a field and a condition for a group to be the multiplicative group of a field, Proc. London Math. Soc., 18, 1968, p. 114-124
 
you can find the following theorem:

> Let $A$ be an abelian group (written multiplicatively). Adjoin a new element $0$ to get the set $F$. Assume that there is a function $f : F \to F$ such that:

> 1) $f(0) = 1$
 
> 2) $f(f(x))=x$
 
> 3) $f(f(x) f(y)) = y f(x f(y^{-1}))$
 
> Then $F$ can be made into a field such that $A = F^*$ and $f(x) = 1 - x$.