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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. 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 $f(x) = 1 - x$.