  The Alesker-Poincare pairing for smooth valuations on manifolds is expressed
in terms of the Rumin differential operator acting on the cosphere-bundle. It
is shown that the derivation operator, the signature operator and the Laplace
operator acting on smooth valuations are formally self-adjoint with respect to
this pairing. As an application, the product structure of the space of SU(2)-
and translation invariant valuations on the quaternionic line is described. The
principal kinematic formula on the quaternionic line is stated and proved.
