  We discuss two surprising properties of a family of polynomials that
generalize the Mahonian $q$-Catalan polynomials, and more generally the
$q$-Schr\"oder polynomials. By interpreting them as
$\mathfrak{sl}_2$-characters, we show that the rational $q$-Schr\"oder
polynomials are parity-unimodal, which means that the even- and odd-degree
coefficients are separately unimodal. Second, we show that they exhibit a
$q=-1$ phenomenon. This is a special case of a more general cyclic sieving
phenomenon for certain transitive $S_n$-actions, deduced from Molien's formula.
