<p>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 reorientations of it. 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 works of Gioan–Las Vergnas and Ding. Some implications in oriented matroid programming and oriented matroid triangulations are also discussed.</p>