Hopf-dihedral (co)homology and $L$-theory

We develop an appropriate dihedral extension of the Connes–Moscovici characteristic map for Hopf $\ast$-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 $\ast$-algebra that carry a Hopf symmetry over a Hopf $\ast$-algebra. Using our machinery we detect a previously unknown $L$-class of the standard Podleś sphere.