The spin reorientation (SR) phenomenon of the square-lattice antiferromagnets RMnAsO (R = Ce, Nd) was investigated by analyzing the spin-exchange interactions between the rare-earth and the transition-metal ions (R3+ and Mn2+, respectively) on the basis of density functional calculations. It is found that the symmetry and strength of the Dzyaloshinskii-Moriya (DM) interaction are determined primarily by the partially filled 4f states of the R3+ ions and that the DM and biquadratic (BQ) exchanges between the R3+ and the Mn2+ ions are unusually strong and control the observed spin reorientation phenomenon. Below their SR temperature, the Mn2+ and Ce3+ moments are orthogonal in CeMnAsO but are collinear in NdMnAsO, because the DM interaction dominates over the BQ interaction for CeMnAsO, while the opposite is the case for NdMnAsO. Experiments designed to test the implications of our findings are proposed.