  Alesker has introduced the space $\mathcal V^\infty(M)$ of {\it smooth
valuations} on a smooth manifold $M$, and shown that it admits a natural
commutative multiplication. Although Alesker's original construction is highly
technical, from a moral perspective this product is simply an artifact of the
operation of intersection of two sets. Subsequently Alesker and Bernig gave an
expression for the product in terms of differential forms. We show how the
Alesker-Bernig formula arises naturally from the intersection interpretation,
and apply this insight to give a new formula for the product of a general
valuation with a valuation that is expressed in terms of intersections with a
sufficiently rich family of smooth polyhedra.
