  In this paper we study the cyclic cohomology of certain x-Hopf algebras:
universal enveloping algebras, quantum algebraic tori, the Connes-Moscovici
x-Hopf algebroids and the Kadison bialgebroids. Introducing their stable anti
Yetter-Drinfeld modules and cocyclic modules, we compute their cyclic
cohomology. Furthermore, we provide a pairing for the cyclic cohomology of
x-Hopf algebras which generalizes the Connes-Moscovici characteristic map to
x-Hopf algebras. This enables us to transfer the x-Hopf algebra cyclic cocycles
to algebra cyclic cocycles.
