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Based on the effective invariant amplitudes of the transitions$J/\psi\to\gamma X(J^P)$ and $X(J^P)\to\phi\phi$, the spin dependence of the$J/\psi\to\gamma X(J^P)\to\gamma\phi\phi$ decay amplitude is obtained in thecase of the intermediate resonances $X$ with $J^P=0^\pm,1^\pm,2^\pm$. Angulardistributions of photons with the definite linear polarizations relative to theplane spanned by the momentum of initial electron and the total momentum of the$\phi\phi$ pair in the reaction $e^+e^-\to J/\psi\to\gammaX(J^P)\to\gamma\phi\phi$ are calculated. It is shown that the sign of theasymmetry of the distributions of the photons polarized in the above plane andorthogonal to it correlates with the signature $P_X(-1)^{J_X}$ of the $X$resonance with given spin-parity $J_X^{P_X}$ so it may help to establish thisquantum number in a way which does not depend on the specific model of the$X(J^P)\to\phi\phi$ amplitude.

The decay $J/\psi\to\gamma\phi\phi$: spin dependence of amplitude and angular distributions of photons with linear polarizations