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This paper presents "oriented pivoting systems" as an abstract framework forcomplementary pivoting. It gives a unified simple proof that the endpoints ofcomplementary pivoting paths have opposite sign. A special case are the Nashequilibria of a bimatrix game at the ends of Lemke-Howson paths, which haveopposite index. For Euler complexes or "oiks", an orientation is defined whichextends the known concept of oriented abstract simplicial manifolds. Ordered"room partitions" for a family of oriented oiks come in pairs of opposite sign.For an oriented oik of even dimension, this sign property holds also forunordered room partitions. In the case of a two-dimensional oik, these areperfect matchings of an Euler graph, with the sign as defined for Pfaffianorientations of graphs. A near-linear time algorithm is given for the followingproblem: given a graph with an Eulerian orientation with a perfect matching,find another perfect matching of opposite sign. In contrast, the complementarypivoting algorithm for this problem may be exponential.

Oriented Euler Complexes and Signed Perfect Matchings