= Solution
The four allowed two-real-<photon> states are $\eta_c$, $\eta_c(2S)$, $\chi_{c0}$ and $\chi_{c2}$. Two <photons> have $C=+1$, excluding $J/\psi$, $\psi(2S)$ and $h_c$; the <Landau-Yang theorem> excludes $\chi_{c1}$ because $J=1$. Both scalar and pseudoscalar two-<photon> amplitudes exist, so $0^{++}$ and $0^{-+}$ are allowed. The <J psi meson> was discovered first because its $1^{--}$ assignment permits direct, resonantly enhanced single-<photon> formation in $e^+e^-$ and a clean dilepton peak in hadron production. The lighter $\eta_c$ cannot be formed by one <photon>; it needs less favorable production channels and has a broader two-<gluon> decay.
A spin-zero $\eta_c$ at rest has no preferred axis, so $d\Gamma/d\Omega_p$ is isotropic after summing final <spins>. A <J psi meson> formed in unpolarized high-energy $e^+e^-$ has transverse $m=\pm1$ alignment. Its $p\bar p$ distribution can be written
$$
\frac{d\Gamma}{d\cos\theta_p}\propto a(1+\cos^2\theta_p)+b\sin^2\theta_p,\qquad a,b\geq0,
$$
where $a,b$ contain the two independent <helicity amplitudes>. It is generally anisotropic; the approximate massless, helicity-conserving limit has $b\to0$. The tabulated branching fraction does not fix $a/b$. An ensemble equally populated in all three <spin> projections would instead be isotropic: alignment of the production ensemble matters.
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