co.combinatorics - What bijection on permutations corresponds under RS to transpose?

When you conjugate diagrams and apply RSK or other Young tableau bijections, the answer is typically bad, with some rare exceptions. The right way to think of RSK is to think of row length being continuous while columns still integer (see e.g. my "Geometric proof of the hook-length formula" paper and refs therein).

An exception: there is a "hidden symmetry" for LR-coefficients when you conjugate all three diagrams - see Hanlon-Sundaram paper (1992). Once you know this bijection, there are natural connections to RSK as described in the long Pak-Vallejo paper and a followup by Azenhas-Conflitti-Mamede (search the web for a paper and a ppt presentation). Together, these do give a complete description of your involution, but making sense of it might require quite a bit of work.