  Let $P$ and $Q$ be finite partially ordered sets on $[d] = \{1, \ldots, d\}$,
and $\mathcal{O}(P) \subset \mathbb{R}^{d}$ and $\mathcal{O}(Q) \subset
\mathbb{R}^{d}$ their order polytopes. The twinned order polytope of $P$ and
$Q$ is the convex polytope $\Delta(P,-Q) \subset \mathbb{R}^{d}$ which is the
convex hull of $\mathcal{O}(P) \cup (- \mathcal{O}(Q))$. It follows that the
origin of $\mathbb{R}^{d}$ belongs to the interior of $\Delta(P,-Q)$ if and
only if $P$ and $Q$ possess a common linear extension. It will be proved that,
when the origin of $\mathbb{R}^{d}$ belongs to the interior of $\Delta(P,-Q)$,
the toric ideal of $\Delta(P,-Q)$ possesses a quadratic Gr\"obner basis with
respect to a reverse lexicographic order for which the variable corresponding
to the origin is smallest. Thus in particular if $P$ and $Q$ possess a common
linear extension, then the twinned order polytope $\Delta(P,-Q)$ is a normal
Gorenstein Fano polytope.
