We perform a complete study of prompt double J/ψ hadroproduction at leading order in the nonrelativistic-QCD factorization framework by including all possible pairings of the cc ¯ Fock states S 1 [8] 0 , S 3 [1,8] 1 , and P 3 [1,8] J with J=0,1,2 . We find that the S 1 [8] 0 and P 3 [8] J channels of J/ψ and ψ ′ production and the P 3 [1] J and S 3 [8] 1 channels of χ cJ production, which have been overlooked so far, greatly dominate at large invariant masses and rapidity separations of the J/ψ pair, and that their inclusion nearly fills the large gap between previous incomplete predictions within the color-singlet model and the recent measurement by the CMS Collaboration at the CERN LHC, leaving room for next-to-leading-order corrections of typical size.