  We develop an appropriate dihedral extension of the Connes-Moscovici
characteristic map for Hopf *-algebras. We then observe that one can use this
extension together with the dihedral Chern character to detect non-trivial
$L$-theory classes of a *-algebra that carry a Hopf symmetry over a Hopf
*-algebra. Using our machinery we detect a previously unknown $L$-class of the
standard Podle\'s sphere.
