A new theoretical analysis explores the spin and parity ($J^P$) quantum number distribution of compact tetraquarks, exotic particles composed of two quarks and two antiquarks. The study, motivated by recent experimental observations of fully charmed tetraquark candidates such as the $X(6600)$, $X(6900)$, and $X(7100)$, suggests that low-energy states of these compact tetraquarks are highly likely to possess a $J^P=2^+$.

The research employs restricted representations of the permutation group $S_4$ to derive the nodal structure of the $qq\bar q\bar q$ system from that of the $qqqq$ system, considering tetrahedral or square configurations for the quarks and antiquarks and orbital angular momenta $L \leq 3$. The results indicate that the dominant features of the low-lying tetraquark spectrum are primarily governed by symmetry constraints, rather than by the details of the underlying dynamics. This is supported by two key observations: the symmetry-induced $J^P$ distribution is similar to that obtained for the three-flavor four-quark system, and the peak of this distribution remains unchanged when chromomagnetic interaction (CMI) effects are incorporated.

These findings imply that the fully charmed tetraquark candidates $X(6600)$, $X(6900)$, and $X(7100)$ may occupy relatively low-lying levels in the fully charmed tetraquark spectrum. Furthermore, the study suggests that mechanisms beyond CMI dynamics are likely involved, potentially mitigating or competing with CMI effects in these compact states. The dynamical robustness of symmetry-based classifications in exotic hadron spectroscopy is thus reinforced.