  Witten's formulation of 2+1 gravity is investigated on the nonorientable
three-manifold R x (Klein bottle). The gauge group is taken to consist of all
four components of the 2+1 Poincare group. We analyze in detail several
components of the classical solution space, and we show that from four of the
components one can recover nondegenerate spacetime metrics. In particular, from
one component we recover metrics for which the Klein bottles are spacelike. An
action principle is formulated for bundles satisfying a certain orientation
compatibility property, and the corresponding components of the classical
solution space are promoted into a phase space. Avenues towards quantization
are briefly discussed.
