  The $J/\psi, \psi(3686) \to \Sigma^0 \bar{\Sigma}^{0}$ processes and
subsequent decays are studied using the world's largest $J/\psi$ and
$\psi(3686)$ data samples collected with the BESIII detector. The strong-$CP$
symmetry is tested in the decays of the $\Sigma^0$ hyperons for the first time
by measuring the decay parameters, $\alpha_{\Sigma^0} = -0.0017 \pm 0.0021 \pm
0.0018$ and $\bar{\alpha}_{\Sigma^0} = 0.0021 \pm 0.0020 \pm 0.0022$. The
weak-$CP$ test is performed in the subsequent decays of their daughter
particles $\Lambda$ and $\bar{\Lambda}$. Also for the first time, the
transverse polarizations of the $\Sigma^0$ hyperons in $J/\psi$ and
$\psi(3686)$ decays are observed with opposite directions, and the ratios
between the S-wave and D-wave contributions of the $J/\psi, \psi(3686) \to
\Sigma^0 \bar{\Sigma}^{0}$ decays are obtained. These results are crucial to
understand the decay dynamics of the charmonium states and the production
mechanism of the $\Sigma^0-\bar{\Sigma}^0$ pairs.
