  Structure in the di-$J/\psi$ mass spectrum observed by the LHCb experiment
around 6.9 and 7.2 GeV is interpreted in terms of $J^{PC}=0^{++}$ and $2^{++}$
resonances between a $cc$ diquark and a $\bar c \bar c$ 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 $cc$-$\bar c \bar c$ tetraquark, which we predict at
$6.871\pm 0.025$ GeV. The dip around 6.75 GeV is ascribed to the opening of the
\Swave\ di-$\chi_{c0}$ channel, while the dip around 7.2 GeV could be
correlated with the opening of the di-$\eta_c(2S)$ or $\Xi_{cc} \bar \Xi_{cc}$
channel. The low-mass part of the di-$J/\psi$ structure appears to require a
broad resonance consistent with a predicted $2^{++}(1S)$ state with invariant
mass around $M_{\rm inv} = 6400$ MeV. Implications for $bb \bar b \bar b$
tetraquarks are discussed.
