  This paper is part of a series of articles on noncommutative geometry and
conformal geometry. In this paper, we reformulate the local index formula in
conformal geometry in such a way to take into account of the action of
conformal diffeomorphisms. We also construct and compute a whole new family of
geometric conformal invariants associated with conformal diffeomorphisms. This
includes conformal invariants associated with equivariant characteristic
classes. The approach of this paper involves using various tools from
noncommutative geometry, such as twisted spectral triples and cyclic theory. An
important step is to establish the conformal invariance of the Connes-Chern
character of the conformal Dirac spectral triple of Connes-Moscovici.
Ultimately, however, the main results of the paper are stated in a purely
differential-geometric fashion.
