  A convex body is unconditional if it is symmetric with respect to reflections
in all coordinate hyperplanes. In this paper, we investigate unconditional
lattice polytopes with respect to geometric, combinatorial, and algebraic
properties. In particular, we characterize unconditional reflexive polytopes in
terms of perfect graphs. As a prime example, we study the signed Birkhoff
polytope. Moreover, we derive constructions for Gale-dual pairs of polytopes
and we explicitly describe Gr\"obner bases for unconditional reflexive
polytopes coming from partially ordered sets
