  We define and study a new family of polytopes which are formed as convex
hulls of partial alternating sign matrices. We use machinery developed in the
study of sign matrix polytopes to determine the inequality descriptions, facet
enumerations, and face lattices of these polytopes. We also study partial
permutohedra that we show arise naturally as projections of these polytopes. We
directly prove vertex and facet enumerations and also characterize the face
lattices of partial permutohedra in terms of chains in the Boolean lattice.
