  We study the joint distribution of descents and sign for elements of the
symmetric group and the hyperoctahedral group (Coxeter groups of types $A$ and
$B$). For both groups, this has an application to riffle shuffling: for large
decks of cards the sign is close to random after a single shuffle. In both
groups, we derive generating functions for the Eulerian distribution refined
according to sign, and use them to give two proofs of central limit theorems
for positive and negative Eulerian numbers.
