Duality in Toric Topology

We characterise integral Poincar\'e duality moment-angle complexes $\mathcal{Z}_{\mathcal{K}}$ in combinatorial terms of the Fan-Wang duality of the simplicial complex $\mathcal{K}$, and consequently in algebraic terms of the Gorenstein duality of the Stanley-Reisner ring $\mathbb{Z}[\mathcal{K}]$. We extend Poincar\'e duality results to certain polyhedral products using polyhedral join products of simplicial complexes.