With the large samples of $e^+e^-$ annihilation data collected at the $J/\psi$ and $\psi(3686)$ resonances, BESIII can use polarization and entanglement to study the production and weak decays of hyperon--antihyperon pairs in multidimensional angular analyses. Recent measurements establish transverse polarization in several spin-$1/2$ channels, determine hyperon and antihyperon decay parameters independently, and provide precise tests of direct $\Delta S=1$ CP violation. The program has now been extended to the spin-$3/2$ $\Xi(1530)^0$, for which vector, quadrupole and octupole polarization components and electromagnetic multipole form factors are resolved. The same entangled-spin information also enables a novel determination of the $\Lambda$ electric dipole moment, improving the previous direct bound by about three orders of magnitude.