Bond breaking in C-60-C-60 dimeric units is believed to play an important role in the onset of magnetism in 2D polymeric C-60. On the basis of density-functional theory, the calculations we present here provide further insight into this mechanism through a quantitative characterization of the bond-breaking processes in the isolated dumbbell-shaped C-60 dimer. In particular, the analysis of the calculated potential energy surfaces for the low-lying singlet and triplet states identifies and locates the S-0-T-2 crossing point, which is crucial for the transition to a magnetic state to take place under thermal conditions. These results also suggest a possible new approach to the production of magnetic polymeric C-60.