A Margulis spacetime is a complete affine 3-manifold M with nonsolvable fundamental group. Associated to every Margulis spacetime is a noncompact complete hyperbolic surface S. We show that every Margulis spacetime is orientable, even though S may be nonorientable. We classify Margulis spacetimes when S is homeomorphic to a two-holed cross-surface Σ, that is, the complement of two disjoint disks in RP2. We show that every such manifold is homeomorphic to a solid handlebody of genus 2, and admits a fundamental polyhedron bounded by crooked planes. Furthermore, the deformation space is a bundle of convex four-sided cones over the space of marked hyperbolic structures. The sides of each cone are defined by invariants of the two components of ?Σ and the two orientation-reversing simple curves. The two-holed cross-surface, together with the three-holed sphere, are the only topologies Σ for which the deformation space of complete affine structures is finite-sided.
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