In the 1970's, work of Curtis, Schori, and West proved that if $X$ is a metric space that is the continuous image of the unit interval containing 2 points, then the hyperspace $2^X$ of all nonempty closed subsets of $X$ topologized by the Hausdorff metric is homeomorphic with the Hilbert cube, $\Pi _{i \ge 1}I_i$.
Theorem: If $\alpha :X\to X$ is an involution of $X$ with nowhere dense fixed point set, then the induced involution of $2^X$ is topologically conjugate to $\beta: \Pi _{i \ge 1}I_i\to \Pi _{i \ge 1}I_i$ given by reflecting the odd-indexed coordinate intervals across their mid-points.