A **partial permutation matrix** $\pi$ is one with at most one 1 in any row and column (the rest 0s). Given one, we can cross out to the East and South (but not Southeast) of each 1. Some boxes get crossed out just horizontally, just vertically, neither, or both (such as the $1$s, which we think of as crossing themselves out both ways). Call these the **SE hooks** of $\pi$.

Similarly, we can define the **NW hooks** of $\rho$. Call $(\pi,\rho)$ **complementary** if when the SE hooks of $\pi$ are overlaid on the NW hooks of $\rho$, each box is crossed out,
and no $1$ crosses out any other. It is easy to see that for any $\pi$, there exists a unique complementary $\rho$, and $\pi \mapsto w_0 \rho w_0$ is an involution.

Has this involution on partial permutations been studied before?

For me, it has come up in studying conormal varieties to Schubert varieties.

ADDED: Here's a familiar subcase. The **diagram** of $\pi$ is the set of boxes not crossed out in its SE hooks. If this is a partition in the NW corner, $\pi$ is called **dominant**, and every partition arises from a unique $\pi$. In this case, $w_0 \rho w_0$ is also dominant, and its diagram is (the rotation of) the complementary partition.