We study in detail the pattern of anomaly cancellation in D = 6 Type IIB Z(N) orientifolds, occurring through a generalized Green-Schwarz mechanism involving several RR antisymmetric tensors and scalars fields. The starting point is a direct string theory computation of the inflow of anomaly arising from magnetic interaction of D-branes, O-planes and fixed points, which are encoded in topological one-loop partition functions in the RR odd spin-structure. All the RR anomalous couplings of these objects are then obtained by factorization. They are responsible for a spontaneous breaking of U(1) actors through a Higgs mechanism involving the corresponding hypermultiplets, Some of them are also related by supersymmetry to gauge couplings involving the NSNS scalars sitting in the tensor multiplets. We also comment on the possible occurrence of tensionless strings when these couplings diverge. (C) 2000 Elsevier Science B.V. All rights reserved.