  The order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$
associated to a partially ordered set $P$ are studied. In this paper, we
introduce the convex polytope $\Gamma(\mathcal{O}(P), -\mathcal{C}(Q))$ which
is the convex hull of $\mathcal{O}(P) \cup (-\mathcal{C}(Q))$, where both $P$
and $Q$ are partially ordered sets with $|P|=|Q|=d$. It will be shown that
$\Gamma(\mathcal{O}(P), -\mathcal{C}(Q))$ is a normal and Gorenstein Fano
polytope by using the theory of reverse lexicographic squarefree initial ideals
of toric ideals.
