Surface diffusion of a water monomer is not as simple as previously imagined. Using state-of-the-art density functional theory, we have obtained important molecular insights relating to the elementary steps of atop-to-atop diffusion of a water monomer. We provide theoretical evidence for an anisotropic effect of rotation-jump coupling on Pd{100}; the preference of H-down tumbling motion along the nearest-neighbor direction of the square lattice was identified. The tumbling motion is attributed to quantum-tunneling-assisted diffusion, while the next-neighbor motion on the square lattice as well as the nearest-neighbor motion on Pd{111} favors molecular-axis-guided classical hopping motion whenever possible. The physical origin of the classical diffusions is discussed in the framework of the electronic structure. Our study gives useful direction for further studies on molecular couplings in the elementary steps.