Top to random and reverse: Analysis of a new descent algebra shuffle

We introduce a new card shuffling operator we call "top-to-random and reverse" derived from the well-known top-to-random shuffle. We show this operator is diagonalizable over Q and describe its eigenvalues and minimal polynomial explicitly. The antipode of our operator lies in Solomon's descent algebra which in turn is anti-isomorphic to a subalgebra of Bidigare's algebra. We introduce both structures in this work, and make ample use of them in the proof our result.