X-ray photoelectron diffraction reveals that C-60 molecules adsorbed in a monolayer on Ag(111) take different orientations depending on the level of potassium doping. For half filling (K3C60) and complete filling (K6C60) of the lowest unoccupied molecular orbital, two distinct orientations are identified, exposing a hexagon or a 6-6 bond to the surface, respectively. For all the intermediate doping levels, the two orientations coexist, which is consistent with phase separation. Characteristic changes in the density of states, calculated for the two orientations within density functional theory, suggest intermolecular electron hopping as a driving force for the molecular reorientation.