  Motivated by recent experimental evidence for apparent $cc\bar c \bar c$
states at LHCb, CMS and ATLAS, we consider how the mass spectrum and decays of
such states can be used to discriminate among their possible theoretical
interpretations, with a particular focus on identifying whether quarks or
diquarks are the most relevant degrees of freedom. Our preferred scenario is
that $X(6600)$ and its apparent partner state $X(6400)$ are the tensor
$(2^{++})$ and scalar $(0^{++})$ states of an S-wave multiplet of $cc\bar c
\bar c$ states. Using tetraquark mass relations which are independent of (or
only weakly dependent on) model parameters, we give predictions for the masses
of additional partner states with axial and scalar quantum numbers.
Additionally, we give predictions for relations among decay branching fractions
to $J/\psi J/\psi$, $J/\psi \eta_c$, $\eta_c\eta_c$ and $D^{(*)} \bar{D}^{(*)}$
channels. The scenario we consider is consistent with existing experimental
data on $J/\psi J/\psi$, and our predictions for partner states and their
decays can be confronted with future experimental data, to discriminate between
quark and diquark models.
