  The Birkhoff polytope, defined to be the convex hull of $n\times n$
permutation matrices, is a well studied polytope in the context of the Ehrhart
theory. This polytope is known to have many desirable properties, such as the
Gorenstein property and existence of regular, unimodular triangulations. In
this paper, we study analogues of the Birkhoff polytope for finite irreducible
Coxeter groups of other types. We focus on a type-$B$ Birkhoff polytope
$\mathcal{BB}(n)$ arising from signed permutation matrices and prove that it
and its dual polytope are reflexive, and hence Gorenstein, and also possess
regular, unimodular triangulations. Noting that our triangulation proofs do not
rely on the combinatorial structure of $\mathcal{BB}(n)$, we define the notion
of an orthant-lattice property polytope and use this to prove more general
results for the existence of regular, unimodular triangulations and unimodular
covers for a significant family of reflexive polytopes. We conclude by
remarking on some connections to Gale-duality, Birkhoff polytopes of other
types, and possible applications of orthant-lattice property.
