We present a complete string theory analysis of all mixed gauge, gravitational and target-space anomalies potentially arising in the simplest heterotic Z(N) orbifold models, with N odd and standard embedding. These anomalies turn out to be encoded in an elliptic index, which can be easily computed; they are found to cancel through a universal GS mechanism induced by the dilaton multiplet. The target-space symmetry is then shown to have a nice geometric interpretation in terms of torsion, and the target-space dependence of the four-dimensional GS couplings can be alternatively rederived from the implicit torsion dependence of the standard ten-dimensional GS couplings. The result is universal and consists essentially of a Bianchi identity for the NSNS B field depending on all the curvatures, and in particular on the target-space curvature.