  We study strong decays of the possible fully-charm tetraquarks recently
observed by LHCb, and calculate their relative branching ratios through the
Fierz rearrangement. Together with our previous QCD sum rule study [Phys. Lett.
B 773, 247 (2017)], our results suggest that the broad structure around
$6.2$-$6.8$ GeV can be interpreted as an $S$-wave $cc\bar c \bar c$ tetraquark
state with $J^{PC} = 0^{++}$ or $2^{++}$, and the narrow structure around 6.9
GeV can be interpreted as a $P$-wave one with $J^{PC} = 0^{-+}$ or $1^{-+}$.
These structures were observed in the di-$J/\psi$ invariant mass spectrum, and
we propose to confirm them in the di-$\eta_c$, $J/\psi h_c$, $\eta_c
\chi_{c0}$, and $\eta_c \chi_{c1}$ channels. We also propose to search for
their partner states having the negative charge-conjugation parity in the
$J/\psi \eta_c$, $J/\psi \chi_{c0}$, $J/\psi \chi_{c1}$, and $\eta_c h_c$
channels.
