  A relationship between signed Eulerian polynomials and the classical Eulerian
polynomials on $\mathfrak{S}_n$ was given by D\'{e}sarm\'{e}nien and Foata in
1992, and a refined version, called signed Euler-Mahonian identity, together
with a bijective proof were proposed by Wachs in the same year. By generalizing
this bijection, in this paper we extend the above results to the Coxeter groups
of types $B_n$, $D_n$, and the complex reflection group $G(r,1,n)$, where the
`sign' is taken to be any one-dimensional character. Some obtained identities
can be further restricted on some particular set of permutations. We also
derive some new interesting sign-balance polynomials for types $B_n$ and $D_n$.
