Theoretical estimates of reactive cross sections for O(D-1)+H-2(X,upsilon =0,j)--> OH(X)+H(S-2), with H-2 rotational quantum numbers j=0 and 1, are obtained for a range of collision energies, E-col. Crossed molecular beam measurements are also used to infer the ratio, r(1,0), of the j=1 and 0 cross sections at E-col=0.056 eV. The theory indicates that the 1 (1)A' potential surface is the most important one. However, the 2 (1)A' and 1 (1)A' surfaces can also contribute. Adiabatic dynamics on the 1 (1)A' surface, particularly at E-col above its 0.1 eV barrier to reaction plays a role. The 2 (1)A' surface, while not correlating with ground electronic state products, can still lead to products via nonadiabatic interactions with the 1 (1)A' surface. Many quantum dynamics and quasiclassical classical trajectory calculations are carried out. Accurate, ab initio based potential energy surfaces are employed. Quantum cross sections are based on helicity decoupled wave packet calculations for several values of total angular momentum. Nonadiabatic wave packet and trajectory surface hopping calculations, where appropriate, are carried out. An interesting, subtle picture emerges regarding the energy dependence of r(1,0). The theoretical results indicate, somewhat surprisingly, that, for E-col<0.1 eV,r(1,0) can be less than unity owing to the anisotropy of the ground state potential. Electronically excited states and nonadiabatic effects contribute to the overall cross sections for E-col>0.1 eV, but the full r(1,0) is only weakly sensitive to excited states. Our experimentally inferred r(1,0) at E-col=0.056 eV, 0.95 +/-0.02, is in quantitative agreement with our best calculation, which suggests that the effect of potential anisotropy is correctly described by theory. The relation between these results and previous experimental findings is discussed.