  We demonstrate that the recent studies of J/psi-pair production by CMS at the
LHC and by D0 at the Tevatron reveal the presence of different production
mechanisms in different kinematical regions. We find out that
next-to-leading-order single parton scattering contributions at alpha_s^5
dominate the yield at large transverse momenta of the pair. Our analysis
further emphasises the importance of double parton scatterings --which are
expected to dominate the yield at large J/psi-rapidity differences-- at large
invariant masses of the pair in the CMS acceptance, and thereby solve a large
discrepancy between the theory and the CMS data. In addition, we provide the
first exact --gauge-invariant and infrared-safe-- evaluation of a class of
leading-P_T (P_T^-4) next-to-next-to-leading-order contributions at alpha_s^6,
which can be relevant in the region of large values of P_T,min=min(P_T1,P_T2).
Finally, we derive simple relations for the feed-down fractions from the
production of an excited charmonium state with a J/psi in the case of the
dominance of the double parton scatterings, which significantly deviate from
those for single parton scatterings. Such relations can be used to discriminate
these extreme scenari, either DPS or SPS dominance.
