Properties of pairs of matrices with flipped kernels and images

Question

In the process of dealing with a certain mathematical question I have found myself dealing with pairs of (square) matrices $(A,B)$ such that $\ker(A) = \operatorname{im(B)}$ and $\ker(B) = \operatorname{im}(A)$. I am curious if there is any established theory on such classes of matrices; and in particular if given a matrix of the form $A + X$ such that $A$ is a before, and the whole matrix $A+X$ is invertible, whether there is a way of finding a more explicit expression for the inverse by using the property some expression using the matrix $B$ in the pair?