  We associate to each infinite primitive Lie pseudogroup a Hopf algebra of
`transverse symmetries', by refining a procedure due to Connes and the first
author in the case of the general pseudogroup. The affiliated Hopf algebra can
be viewed as a `quantum group' counterpart of the infinite-dimensional
primitive Lie algebra of the pseudogroup. It is first constructed via its
action on the \'etale groupoid associated to the pseudogroup, and then realized
as a bicrossed product of a universal enveloping algebra by a Hopf algebra of
regular functions on a formal group. The bicrossed product structure allows to
express its Hopf cyclic cohomology in terms of a bicocyclic bicomplex analogous
to the Chevalley-Eilenberg complex. As an application, we compute the relative
Hopf cyclic cohomology modulo the linear isotropy for the Hopf algebra of the
general pseudogroup, and find explicit cocycle representatives for the
universal Chern classes in Hopf cyclic cohomology. As another application, we
determine all Hopf cyclic cohomology groups for the Hopf algebra associated to
the pseudogroup of local diffeomorphisms of the line.
