Quasiclassical calculations have been carried out near threshold for the reaction of O + H-2(upsilon = 0, 1, 2; J) --> OH + H for J = 0 to 10 to gain an understanding of the effect of reactant rotation and vibration on this reaction. The calculations were done on the Johnson-Winter potential energy surface (PES); this surface has a barrier to reaction which is lowest in the linear configuration (chi = 180 degrees). For values of r, the H-H bond length, less than 1.0 Angstrom, the PES is fairly prolate in nature. (In the classification of Levine, a surface is prolate if for a given distance R(CM) from O to the center of mass of H-2, the potential is higher for the collinear configuration than for the perpendicular configuration (chi = 90 degrees). The surface is oblate if the reverse is true.) For the reaction of O + H-2(upsilon = 0) the reactive cross section Q(R) increases monotonically with J. Our calculations demonstrate that the prolate PES ''disorients'' the O-H-H angle away from linear, and reduces Q(R) substantially for J = 0. An increase in J reduces the effect of this disorientation and so increases Q(R). The reaction with O is much different for H-2 (upsilon = 1, 2). Now the average value of r is long enough that the PES is oblate during the last part of the reaction. Near threshold for both vibrational levels Q(R) rises from J = 0 to 2, then declines to a minimum at J = 5, and finally rises monotonically to J = 10. The decline has been seen for other reactions which take place on oblate surfaces such as F + H-2. The explanation of the decline is that for a rotating target the O atom approaches the reactant valley obliquely, and many trajectories bounce off the repulsive potential wall near chi = 90 degrees rather than react. However, for J values above 5 the H-2 molecule has enough rotational energy to rotate through the barrier at chi = 90 degrees rather than bounce off it, so the collision continues on to reaction. This causes Q(R) to rise. The results obtained here are compared with the quantal calculations by Chatfield et al. on this system. A number of interesting similarities are observed.