D. Kastler DUAL PAIRS OF QUANTUM SPACES (35K, TeX) ABSTRACT. We discuss the notion of ``dual riemannian quantum space'' corresponding to the ``algebraic condition'' in Alain Connes' definition of non-commutative Poincar\'e duality. We show that the DeRham complex of a dual riemannian quantum space is the homomorphic image of a skew tensor product, this leading to a natural definition of ``biconnection''.