ABSTRACT. A definition of PL transversality is given, using the orderreversing duality on partially ordered sets. David Stone's theory of stratified polyhedra is thereby simplified; in particular, the symmetry of blocktransversality is proved. Also, polyhedra satisfying Poincare' duality are characterized geometrically.The purpose of this paper is to develop a simple theory of transversality for polyhedra in piecewise linear manifolds. Our main tool is a canonical geometric duality for "structured" cone complexes. Cone complexes have also been used by M. Cohen [7] and E. Akin [2] to study simplicial maps. The rigid geometry provided by a structured cone complex on a polyhedron is analogous to a Riemannian metric on a smooth manifold (as manifested by its relation to duality and transversality). In fact, structured cone complexes are equivalent to van Kampen's combinatorial "star complexes" [10] (1929 [22]. The stumbling block of Stone's stratified blocktransversality theory was the question of whether transversality of polyhedra in a manifold is symmetric. Furthermore, symmetry of blocktransversality easily implies that Stone's definition is equivalent to Rourke and Sanderson's "mocktransversality" for polyhedrayielding a unified and versatile theory. Our analysis of transversality stems from its close relation to Poincaré duality-in fact our definition echoes Lefschetz' classical (1930) definition of transversality of cycles in a combinatorial manifold [12]. If X is a closed PL manifold, and C is a cell complex on X, a choice of cone structures for the cells of C determines a dual cell complex C*. We define polyhedra P and Q to be transverse in A"" if there is such a structured cell complex C on X with P a subcomplex of C and Q a subcomplex of C*. This definition is symmetric, and our main result is that it is equivalent to Stone's definition.