  We construct a variant $\mathcal{K}_n$ of the Hopf algebra $\mathcal{H}_n$,
which acts directly on the noncommutative model for the generic space of leaves
rather than on its frame bundle. We prove that the Hopf cyclic cohomology of
$\mathcal{K}_n$ is isomorphic to that of the pair $(\mathcal{H}_n, {\mathop{\rm
GL}_n})$ and thus consists of the universal Hopf cyclic classes. We then
realize these classes in terms of geometric cocycles.
