In October 2010, I published a Monthly problem that introduced the concept of a fair permutation, which is a permutation $\pi$ such that for every $i$, either $\pi(i) > i$ and $\pi^{-1}(i) > i$, or $\pi(i) \le i$ and $\pi^{-1}(i) \le i$. Equivalently, every cycle of $\pi$ is either a fixed point or an alternating cycle of even length, meaning that if $z$ is the largest element of the cycle, then $$z > \pi(z) < \pi(\pi(z)) > \pi(\pi(\pi(z))) < \cdots < z.$$

I don't think anyone has studied fair permutations explicitly before. However, a permutation $\sigma$ on $\lbrace1, 2, \ldots, 2m\rbrace$ is said to be a Salié permutation if for some $r\le m$, $$\sigma(1) < \sigma(2) > \sigma(3) < \cdots < \sigma(2r)$$ and $$\sigma(2r) < \sigma(2r+1) < \sigma(2r+2) < \cdots < \sigma(2m).$$ These have been studied before, and one can show by generatingfunctionology that the number of fair permutations is twice the number of Salié permutations. This is very suggestive and hints at a close connection.

Question: Can one construct an explicit 2-to-1 map from fair permutations to Salié permutations?