  We study symmetries enjoyed by the polynomials enumerating non-degenerate
flags in finite vector spaces, equipped with a non-degenerate alternating
bilinear, hermitian or quadratic form. To this end we introduce Igusa-type
rational functions encoding these polynomials and prove that they satisfy
certain functional equations. Some of our results are achieved by expressing
the polynomials in question in terms of what we call parabolic length functions
on Coxeter groups of type $A$. While our treatment of the orthogonal case
exploits combinatorial properties of integer compositions and their
refinements, we formulate a precise conjecture how in this situation, too, the
polynomials may be described in terms of parabolic length functions.
