  Stanley introduced two classes of lattice polytopes associated to posets,
which are called the order polytope ${\mathcal O}_P$ and the chain polytope
${\mathcal C}_P$ of a poset $P$. It is known that, given a poset $P$, the
Ehrhart polynomials of ${\mathcal O}_P$ and ${\mathcal C}_P$ are equal to the
order polynomial of $P$ that counts the $P$-partitions. In this paper, we
introduce the enriched order polytope of a poset $P$ and show that it is a
reflexive polytope whose Ehrhart polynomial is equal to that of the enriched
chain polytope of $P$ and the left enriched order polynomial of $P$ that counts
the left enriched $P$-partitions, by using the theory of Gr\"{o}bner bases. The
toric rings of enriched order polytopes are called enriched Hibi rings. It
turns out that enriched Hibi rings are normal, Gorenstein, and Koszul. The
above result implies the existence of a bijection between the lattice points in
the dilations of ${\mathcal O}^{(e)}_P$ and ${\mathcal C}^{(e)}_P$. Towards
such a bijection, we give the facet representations of enriched order and chain
polytopes.
