I would like to have a list of pairs (or tuples) of combinatorial statistics that are (known or conjectured) to have symmetric distribution. Ideally, something like this has already been compiled, otherwise, maybe this is the place to do so.

Some examples:

on Dyck paths: area and bounce, returns to the axis and length of the last descent

on permutations: major index and number of inversions

on perfect matchings, set partitions and permutations: crossings and nestings, the maximal crossing number and the maximal nesting number

Maybe it's best to have one family of objects per answer. Edit: originally, I had only *joint* symmetric distribution in mind. However, lists of equidistributed tuples are also very good to have. Please indicate in your answer what your tuple satisfies!

Definitions:

Statistics $stat_1,stat_2,\dots,stat_n$ on a set $X$ are *equidistributed* if
$$\sum_{x\in X}q^{stat_1(x)} = \sum_{x\in X}q^{stat_2(x)} = \dots \sum_{x\in X}q^{stat_n(x)}.$$

A tuple of statistics $(stat_1,stat_2,\dots,stat_n)$ on a set $X$ has a *symmetric* distribution if its generating function
$$F_{stat_1,stat_2}(q,t) := \sum_{x\in X}x_1^{stat_1(x)}x_2^{stat_2(x)}\dots x_n^{stat_n(x)}$$
is symmetric in $x_1,x_2,\dots,x_n$.