Let $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \}$ be the unit disk.

We say a function $f : \mathbb{D} \rightarrow \mathbb{D}$ is a **winner** if it satisfies the following:

1) it is a homeomorphism

2) if $f(x) = y$, then $f(y) = x$

3) the only fixed point is the origin $(0,0)$

It is easy to give at least one example of a winner function: the "point reflection" centered at the origin (i.e., a 180-degree rotation). Are there other examples?

As pointed out by Bill Johnson (see below), winners are invariant under conjugation by origin-preserving homeomorphisms. This raises the following question: is there another "conjugacy class" of winners?