  Using $1310.6\times10^6$ $J/\psi$ and $447.9\times10^6$ $\psi(3686)$ events
collected with the BESIII detector at the BEPCII $e^{+}e^{-}$ collider, the
branching fractions and the angular distributions of $J/\psi$ and $\psi(3686)$
decays to $\Lambda\bar{\Lambda}$ and $\Sigma^0\bar{\Sigma}^0$ final states are
measured. The branching fractions are in agreement with, and much more precise
than, the averages of previously published results. The polar angular
distributions of $\psi(3686)$ decays are measured for the first time, while
those of $J/\psi$ decays are measured with much improved precision. In
addition, the ratios of branching fractions
$\frac{\mathcal{B}(\psi(3686)\to\Lambda\bar{\Lambda})}{\mathcal{B}(J/\psi\to\Lambda\bar{\Lambda})}$
and
$\frac{\mathcal{B}(\psi(3686)\to\Sigma^0\bar{\Sigma}^0)}{\mathcal{B}(J/\psi\to\Sigma^0\bar{\Sigma}^0)}$
are determined to test the "12\% rule".
