The BESIII experiment at the electron-positron collider BEPCII in Beijing has collected the world's largest data at the $J/\psi$ and $\psi(2S)$ resonances. The fact that the hyperons are produced with a non-zero spin polarization that is straightforward to parameterize in processes involving virtual photons or vector mesons, enable direct and precise CP tests. These CP-symmetry tests can be performed on e.g. $e^+e^-\to J/\psi \to \Lambda \overline{\Lambda} \to p \pi^- \overline{p} \pi^+$, $e^+e^-\to J/\psi \to \Sigma^+ \overline{\Sigma}^- \to p \pi^0 \overline{p} \pi^0$ or $e^+e^-\to J/\psi \to \Xi^- \overline{\Xi}^+ \to \Lambda \pi^- \overline{\Lambda} \pi^+ \to p \pi^- \pi^- \overline{p} \pi^+ \pi^+$. For the $\Xi^- \to \Lambda \pi^-$ decay the exclusive measurement of the final state particles allows for three independent CP tests and the determination of the strong and weak phase differences. The weak phase difference is the first measurement for any baryon decay. The decay parameter $\phi_{\Xi}$ is measured with a precision similar to the hitherto most precise measurement, performed by the HyperCP experiment at Fermilab. This is despite the fact that the BESIII data sample is three orders of magnitude smaller than that of HyperCP.