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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 theconvex hull of $\mathcal{O}(P) \cup (- \mathcal{O}(Q))$. It follows that theorigin of $\mathbb{R}^{d}$ belongs to the interior of $\Delta(P,-Q)$ if andonly 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 withrespect to a reverse lexicographic order for which the variable correspondingto the origin is smallest. Thus in particular if $P$ and $Q$ possess a commonlinear extension, then the twinned order polytope $\Delta(P,-Q)$ is a normalGorenstein Fano polytope.

Quadratic Gr\"obner bases of twinned order polytopes