  Extending the notion of geometric bijections for regular matroids, introduced
by the first and third author with Matthew Baker, we describe a family of
bijections between bases of an oriented matroid and special orientations. These
bijections are specified by a pair of circuit and cocircuit signatures coming
respectively from a generic single-element lifting and extension. We then
characterize generic single-element liftings and extensions using these
bijections. We also explain the relation of our work with the works of
Gioan--Las Vergnas and Ding. Some implications in oriented matroid programming
and oriented matroid triangulations are also discussed.
