  The $J/\psi \to \Xi^0 \bar{\Xi}^{0}$ process and subsequent decays are
investigated using $(10087 \pm 44)\times 10^6$ $J/\psi$ events collected at the
BESIII experiment. The decay parameters of $\Xi^0$ and $\bar{\Xi}^0$ are
measured with greatly improved precision over previous measurements to be
$\alpha_{\Xi} = -0.3750 \pm 0.0034 \pm 0.0016$, $\bar{\alpha}_{\Xi} = 0.3790
\pm 0.0034 \pm 0.0021$, $\phi_{\Xi} = 0.0051 \pm 0.0096 \pm 0.0018$~rad,
$\bar{\phi}_{\Xi} = -0.0053 \pm 0.0097 \pm 0.0019$~rad, where the first and the
second uncertainties are statistical and systematic, respectively. From these
measurements, precise $CP$ symmetry tests in $\Xi^0$ decay are performed, and
$A^{\Xi}_{CP} = (-5.4 \pm 6.5 \pm 3.1) \times 10^{-3}$ and
$\Delta\phi^{\Xi}_{CP} = (-0.1 \pm 6.9 \pm 0.9) \times 10^{-3}$~rad are
consistent with $CP$ conservation. The sequential decay also enables a
separation of weak and strong phase differences, which are found for the first
time to be $\xi_{P}-\xi_{S} = (0.0 \pm 1.7 \pm 0.2) \times 10^{-2}$~rad and
$\delta_{P}-\delta_{S} = (-1.3 \pm 1.7 \pm 0.4)\times 10^{-2}$~rad,
respectively. In addition, we measure the $\Lambda$ decay parameters and test
$CP$ symmetry in $\Lambda$ decays.
