This paper examines spinor structures and two-component spinor fields in in a class of spacetimes that are spacedrientable but not time-orientable. The space-oriented frames form a principal bundle acted on by the group of p r o p nononhochronous Lomtz transformations. This p u p has two double coverings, Sint and Sin-, but only Sinacts on the usual two-component spinors associated wilh Weyl neutrinos in Minkowski space Consideration is initially reshicled to Lorentzian universes-from-nothing. geometries, like antipodally identified deSiner space, that have a single spacelike boundary and a smooth meuic wilh Lorentzian signalure. Every such spacetime has a Sin+ S t " r e , but only a subclass has a Sinstructure. Inequivalent Sin+-and Sin--spinor structures correspond to members of two classes of homomorphisms from XI(=) to &, where ; i i is the orientable double covering of the spacetime m i f o l d M. For general time-nonorientable spacetimes. a similar classification is obtained of Sin' sWcNres in terms of homomorphisms from q ( E ) to & where E is the bundle of space-oriented frames of M .

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