In this paper, we introduce a convenient, geometric parameterization of the space of non-orthogonal primal-dual triangulations. Motivated by practical numerical issues in a number of modeling and simulation problems, we relax the conventional orthogonality of mesh duality and introduce the weaker notion of a compatible dual complex to a primal triangulation. We show that the existence of compatible dual complexes is guaranteed only for a particular type of triangulation known as weakly regular. We provide a simple parameterization of the entire space of compatible pairs of primal triangulations and dual complexes that extends the well known weighted Delaunay / power diagram duality, and allows us to easily explore a space of primal-dual structures much larger than the space of orthogonal primal-dual structures. This parameterization may play an important role in discrete optimization problems such as optimal mesh generation.