  We introduce a cup-cap duality in the Koszul calculus of N-homogeneous
algebras. As an application, we prove that the graded symmetry of the Koszul
cap product is a consequence of the graded commutativity of the Koszul cup
product. We propose a conceptual approach that may lead to a proof of the
graded commutativity, based on derived categories in the framework of DG
algebras and DG bimodules. Various enriched structures are developed in a
weaker situation corresponding to N>2.
