Final rotational state J distributions of oriented NO molecules scattered with a translational energy E(i) of 0.1 and 0.34 eV from Ag(111) are directly measured. When the O end is preferentially directed to the surface, the average rotational excitation is more than 50% higher than when the N end is preferentially directed to the surface, at E(i) = 0.34 eV. For scattering closer to the surface than specular, a rotational rainbow is observed around J = 23.5. The corresponding average initial orientation angle gamma-s of these molecules is determined to be (47 +/- 8)-degrees. This value is in accordance with that found in theoretical studies by Voges and Schinke, who suppose that the rotational excitation is related to orientational anisotropy in the repulsive part of the potential. In the N-end distribution a single peak is found with its maximum around J = 8.5, which is also observed in the O-end distribution. For angles closer to the normal than specular, the rotational rainbow disappears in the O-end distribution. The N-end distribution, however, does not change. The shift between the angular density distributions for the O end and N end can be only partially explained by the difference in rotational excitation between the two ends of the molecule. Other mechanisms result in transfer of parallel momentum to other degrees of freedom, especially for N-end collisions. Some of these mechanisms, such as corrugation of the surface and multiple bounces, are discussed.