One of the most relevant features of the O(3P) + H2 reaction is that it occurs on two different potential energy surfaces (PESs) of symmetries A' and A'' that correlate reactants and products. The respective saddle points, which correspond to a collinear arrangement, are the same for both PESs, whilst the barrier height rises more abruptly on the 3A' PES than on the 3A'' PES. Accordingly, the reactivity on the 3A'' PES should be always higher than on the 3A' PES. In this work, we present accurate quantum-scattering calculations showing that this is not always the case for rotationless reactants, where dynamical factors near the reaction threshold cause the 3A' PES to dominate at energies around the barrier. Further calculation of cross sections and Λ-doublet populations has allowed us to establish how the reaction mechanism changes from the deep tunneling regime to hyperthermal energies.