  A general setting to study a certain type of formulas, expressing characters
of the symmetric group $\mathfrak{S}_n$ explicitly in terms of descent sets of
combinatorial objects, has been developed by two of the authors. This theory is
further investigated in this paper and extended to the hyperoctahedral group
$B_n$. Key ingredients are a new formula for the irreducible characters of
$B_n$, the signed quasisymmetric functions introduced by Poirier, and a new
family of matrices of Walsh--Hadamard type. Applications include formulas for
natural $B_n$-actions on coinvariant and exterior algebras and on the top
homology of a certain poset in terms of the combinatorics of various classes of
signed permutations, as well as a $B_n$-analogue of an equidistribution theorem
of D\'esarm\'enien and Wachs.
