Dual cell structures on manifolds

This text gives some details explaining that even with triangulations this may not work: a construction of a non-combinatorial triangulation (as double suspension of a homology sphere) and further references.

In "Combinatorial cell complexes and Poincare duality", T. Basak defines a combinatorial cell complex (c.c.c.), which seem to meet your "polyhedral (convex cell) complex" description. For a c.c.c. $S$, the author defines the opposite c.c.c. $S^\circ$ as the reversal of the partial order. This construction of $S^\circ$ appears to yield the "dual polyhedral structure with opposite poset" you desire.