  This note discusses the cyclic cohomology of a left Hopf algebroid
($\times_A$-Hopf algebra) with coefficients in a right module-left comodule,
defined using a straightforward generalisation of the original operators given
by Connes and Moscovici for Hopf algebras. Lie-Rinehart homology is a special
case of this theory. A generalisation of cyclic duality that makes sense for
arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic
homology of an associative algebra provides an example of this dual theory that
uses coefficients that are not necessarily stable anti Yetter-Drinfel'd
modules.
