The proof of the Combinatorial Hard Lefschetz Theorem for the "virtual'' intersection cohomology of a not necessarily rational polytopal fan as presented by Karu completely establishes Stanley's conjectures for the generalized $h$-vector of an arbitrary polytope. The main ingredients, Poincaré Duality and the Hard Lefschetz Theorem, rely on an intersection product. In its original constructions, given independently by Bressler and Lunts on the one hand, and by the authors of the present article on the other, there remained an apparent ambiguity. The recent solution of this problem by Bressler and Lunts uses the formalism of derived categories. The present article instead gives a straightforward approach to combinatorial duality and a natural intersection product, completely within the framework of elementary sheaf theory and commutative algebra, thus avoiding derived categories.