Suppose you want to find the average of an $n\times n$ diagonal matrix $A$ over all possible rotations, $$ \langle A\rangle = \int\limits_\text{SO($n$)} Q^T A Q \; dQ. $$
It's easy enough to do this for $n=2$ because you have only one degree of freedom. I suspect that $n=3$ would be "simple", but tedious.
Using Monte-Carlo, I conjecture that $\langle A\rangle = \text{diag}\left[\text{tr}(A)/n \right]$. But I'm curious if there's a way that you could derive this relation analytically for a general $n$.
I suspect that you could make some symmetry arguments to take the problem of an integral over $\text{SO}(n)$ to a sum over permutation matrices.
Why take $A$ to be diagonal? Because really I'm interested in $A$ being positive-definite, and when that's the case, you can just diagonalize it and then rotate into the principal coordinate-frame.