Density functional theory (DFT) and low-temperature scanning tunneling microscopy (STM) have been combined to examine the bonding of individual C-60 molecules on Cu(111). Energy-resolved differential-conductance maps have been measured for individual C-60 molecules adsorbed on a Cu(111) surface by means of low-temperature STM, which are compared to and complemented by theoretically computed spectral images. It has been found that C-60 chemisorbs with a six-membered ring parallel to the surface at two different Cu(111) binding sites that constitute two exclusive hexagonal sublattices. On each sublattice, C-60 is bonded in one particular rotational conformer, i.e., C-60 molecules bind to the Cu(111) surface in two different azimuthal orientations differing by 60 depending on which sublattice the binding site belongs to. The binding conformation of C60 and its orientation with regard to the copper surface can be deduced by this joint experimental-theoretical approach. Six possible pairs of C60 configurations on three different Cu surface binding sites have been identified that fulfil the requirements of the two sublattices and are consistent with all experimental and theoretical data. Theory proposes that two of these configuration pairs are most likely. We have found that DFT does not get the binding energy between rotational conformers in the correct order. We also report two different C-60 monolayers on Cu(111): one with alternating orientations of neighboring molecules at low temperature and the other with (4 x 4) structure after annealing above room temperature.