The Robinson-Schensted correspondence is a bijection between elements of the symmetric group and ordered pairs of standard tableaux of the same shape.

Some simple operations on tableaux correspond to simple operations on the group: switching the tableaux corresponds to inverse on the group.

What about taking the transpose of the tableaux? Does that correspond to something easily described on permutations?

In order to rule out easy guesses, let me describe this on $S_3$:

- the identity switches with the involution 321.
- the transpositions 213 and 132 switch.
- the 3-cycles 312 and 231 switch.

In general, this operation preserves being order $\leq 2$ (since this is equivalent to the P- and Q-symbols being the same).