Quadratic Gr\"obner bases of twinned order polytopes

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.