  Stanley introduced in 1986 the order polytope and the chain polytope for a
given finite poset. These polytopes contain much information about the poset
and have given rise to important examples in polyhedral geometry. In 1993,
Reiner introduced signed posets as natural type-B analogues of posets. We
define and study signed order and chain polytopes. Our results include
convex-hull and halfspace descriptions, unimodular triangulations, Ehrhart
$h^*$-polynomials and their relations to signed permutation statistics, and a
Gorenstein characterization of signed order and chain polytopes.
