A comprehensive study of all-charm tetraquark candidates is presented in this paper. The search is performed in the $J/\psi J/\psi$ mass spectrum using proton–proton collision data at $\sqrt{s}=13$ and 13.6 TeV collected with the CMS detector at the LHC. With about 3.6 times more $J/\psi J/\psi$ pairs compared to the Run 2 dataset alone, the combined Run 2 (2016--2018) and Run 3 (2022--2024) sample provides significantly improved sensitivity to rare structures. In the $J/\psi J/\psi \rightarrow \mu^{+}\mu^{-}\mu^{+}\mu^{-}$ channel, three structures are observed each with a significance well above $5\sigma$, denoted $X(6600)$, $X(6900)$, and $X(7100)$, consistent with the earlier Run 2 observation. In addition, quantum interferences between these states are observed, each with a significance above $5\sigma$, implying they share the same spin–parity quantum numbers. In addition, a search is conducted in the $J/\psi \psi(2S)\rightarrow \mu^{+}\mu^{-}\mu^{+}\mu^{-}$ channel, revealing two structures with masses and widths consistent with $X(6900)$ and $X(7100)$. To further explore the nature of these states, a spin–parity measurement is conducted with the Run 2 $J/\psi J/\psi$ data. Assuming common quantum numbers for the three states, as motivated by the interference pattern, the results favor a $J^{PC}=2^{++}$ assignment---an unusual quantum number among known hadrons. The combined features of these analyses point to a coherent picture in which the $J/\psi J/\psi$ states observed by CMS constitute a family of all-charm tetraquarks, favoring a diquark-antidiquark structure in which both diquarks are in a spin-1 state and aligned. The findings provide new insights into the internal dynamics of exotic resonances.