Suppose that $M$ is a compact manifold without boundary (smooth if you like), and suppose further that $M$ is equipped with a regular CW-complex structure. Denote the face poset of this CW-complex by $P$.

Is it true that there is a dual cell structure, also a regular CW complex, and whose face poset is the opposite (dual) poset $P^{\mbox{op}}$?

So far I can't find a reference for this in the literature. One often starts with a triangulation of $M$, and I am looking for a more general statement.