Structure in the di-J/ψ mass spectrum observed by the LHCb experiment around 6.9 and 7.2 GeV is interpreted in terms of JPC=0++ and 2++ resonances between a cc diquark and a c¯c¯ antidiquark, using a recently confirmed string-junction picture to calculate tetraquark masses. The main peak around 6.9 GeV is likely dominated by the 0++(2S) state, a radial excitation of the cc-c¯c¯ tetraquark, which we predict at 6.871±0.025 GeV. The dip around 6.75 GeV is ascribed to the opening of the S-wave di-χc0 channel, while the dip around 7.2 GeV could be correlated with the opening of the di-ηc(2S) or ΞccΞ¯cc channel. The low-mass part of the di-J/ψ structure appears to require a broad resonance consistent with a predicted 2++(1S) state with invariant mass around Minv=6400 MeV. Implications for bbb¯b¯ tetraquarks are discussed.