An answer to this question would provide an explicit counterexample to this question, but otherwise I don't know if it is interesting.

Consider all permutations $\pi$ on the natural numbers such that for each $i$, $\pi(3i)=2i$ and $\lbrace \pi(3i+1),\pi(3i+2)\rbrace = \lbrace 4i+1,4i+3\rbrace$.

**The question:** Is there such a permutation for which all the cycles have finite length?

By computer it is easy to find choices that make the cycles containing the first several thousand integers finite.