We have grown ultrathin epitaxial Co films on three low-index surfaces of Pd. Their magnetic properties studied by the magneto-optical Kerr effect are correlated with the surface morphology analyzed by scanning tunneling microscopy. A perpendicular magnetic anisotropy in the films appeared after exposure to residual gas atmosphere at low temperature, after coverage with an Au overlayer or after annealing at 370 K and above. For one- and two-monolayer-thick films the hysteresis loops are reversed with respect to those of thicker Co films. This is due the negative polar Kerr rotation contribution from the Co/Pd interface dominating over the positive polar Kerr rotation contribution from the noninterface part of the Co film. Reversed loops are not seen when the Co films grow in a three-dimensional mode. This indicates a clear correspondence between the mode of growth and the magneto-optical response. Near the film thickness where the Kerr rotation changes sign a remarkable temperature behavior is observed which is discussed as a superposition of two different magneto-optical contributions. The results qualitatively agree with the available ab initio band-structure calculations for the Co/Pd multilayer structures with variable thickness of the Co layers.