  We prove a cyclic cohomological analogue of Haefliger's van Est-type theorem
for the groupoid of germs of diffeomorphisms of a manifold. The differentiable
version of cyclic cohomology is associated to the algebra of transverse
differential operators on that groupoid, which is shown to carry an intrinsic
Hopf algebraic structure. We establish a canonical isomorphism between the
periodic Hopf cyclic cohomology of this extended Hopf algebra and the
Gelfand-Fuchs cohomology of the Lie algebra of formal vector fields. We then
show that this isomorphism can be explicitly implemented at the cochain level,
by a cochain map constructed out of a fixed torsion-free linear connection.
This allows the direct treatment of the index formula for the hypoelliptic
signature operator - representing the diffeomorphism invariant transverse
fundamental $K$-homology class of an oriented manifold - in the general case,
when this operator is constructed by means of an arbitrary coupling connection.
