  This paper presents "oriented pivoting systems" as an abstract framework for
complementary pivoting. It gives a unified simple proof that the endpoints of
complementary pivoting paths have opposite sign. A special case are the Nash
equilibria of a bimatrix game at the ends of Lemke-Howson paths, which have
opposite index. For Euler complexes or "oiks", an orientation is defined which
extends 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 for
unordered room partitions. In the case of a two-dimensional oik, these are
perfect matchings of an Euler graph, with the sign as defined for Pfaffian
orientations of graphs. A near-linear time algorithm is given for the following
problem: given a graph with an Eulerian orientation with a perfect matching,
find another perfect matching of opposite sign. In contrast, the complementary
pivoting algorithm for this problem may be exponential.
