  We pose counting problems related to the various settings for Coxeter-Catalan
combinatorics (noncrossing, nonnesting, clusters, Cambrian). Each problem is to
count "twin" pairs of objects from a corresponding problem in Coxeter-Catalan
combinatorics. We show that the problems all have the same answer, and, for a
given finite Coxeter group W, we call the common solution to these problems the
W-biCatalan number. We compute the W-biCatalan number for all W and take the
first steps in the study of Coxeter-biCatalan combinatorics.
