It's well known that LU decomposition is only numerically stable if it's combined with row and/or column pivoting. It makes me wonder if there are other matrix decompositions that can profitably be combined with pivoting, that are not simply special cases of LDU decomposition with block diagonal D.

More broadly, which matrix decompositions feature permutation matrices?

The motivation for this question is to gain a unified understanding of the features of matrix decompositions.