Additive triples of bijections, or the toroidal semiqueens problem

We prove an asymptotic for the number of additive triples of bijections {1, ..., n} -> Z/nZ, that is, the number of pairs of bijections pi(1), pi(2) : {1, ..., n} -> Z/nZ such that the point-wise sum pi(1) + pi(2) is also a bijection. This problem is equivalent to counting the number of orthomorphisms or complete mappings of Z/nZ, to counting the number of arrangements of n mutually nonattacking semiqueens on an n x n toroidal chessboard, and to counting the number of transversals in a cyclic Latin square. The method of proof is a version of the Hardy-Littlewood circle method from analytic number theory, adapted to the group (Z/nZ)(n).