The random-to-below shuffle of a deck of cards consists of removing any card randomly (with uniform probability), and inserting it anywhere below (with uniform probability). When looking at the eigenvalues of its transition matrix, they all seem to be rational and positive. This is surprising for a non-symmetric matrix, and suggests some combinatorial interpretation. We discuss some ongoing attempts at giving a recursive explanation that involves standard Young tableaux, and makes connection with the well studied top-to-random shuffle. We finally describe a more straightforward way to compute a stopping time for this shuffle, answering the question of how long we would need to shuffle to get a well-mixed deck using this shuffling technique.