Abstract: 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 singleelement 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.

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