  Let $(P,\leq_P)$ and $(Q,\leq_Q)$ be finite partially ordered sets with
$|P|=|Q|=d$, and $\mathcal{C}(P) \subset \mathbb{R}^d$ and $\mathcal{C}(Q)
\subset \mathbb{R}^d$ their chain polytopes. The twinned chain polytope of $P$
and $Q$ is the lattice polytope $\Gamma(\mathcal{C}(P),\mathcal{C}(Q)) \subset
\mathbb{R}^d$ which is the convex hull of $\mathcal{C}(P) \cup
(-\mathcal{C}(Q))$. It is known that twinned chain polytopes are Gorenstein
Fano polytopes with the integer decomposition property. In the present paper,
we study combinatorial properties of twinned chain polytopes. First, we will
give the formula of the volume of twinned chain polytopes in terms of the
underlying partially ordered sets. Second, we will identify the
facet-supporting hyperplanes of twinned chain polytopes in terms of the
underlying partially ordered sets. Finally, we will provide the vertex
representations of the dual polytopes of twinned chain polytopes.
