We analyze joint angular distributions of a charmonium decay to the ΞΞ pair using the Ξ → Λπ → pπ − π weak decay chain for the cascade and the charge conjugated mode for the anticascade. The decays allow a direct comparison of the baryon and antibaryon decay properties and a sensitive test of CP symmetry in the strange baryon sector. We show that all involved decay parameters can be determined separately in vector and (pseudo)scalar charmonia decays into ΞΞ due to the spin correlations between the weak decay chains. Contrary to the recently measured e + e − → J/ψ → ΛΛ process, the transverse polarization of the cascade is not needed and has almost no impact on the uncertainties of the decay parameters.The ongoing experimental studies of the combined charge conjugation parity (CP) symmetry violation in particle decays aim to find effects that are not expected in the Standard Model (SM), such that new dynamics is revealed. The existence of CP violation in kaon and beauty meson decays is well established [1-3]. The first observation of the CP violation for charm mesons was reported this year by the LHCb experiment [4] and in the bottom baryon sector evidence is mounting [5]. All the observations are consistent with the SM expectation. However, no signal is detected in decays of baryons with strange quark(s) (hyperons). Hyperon decays offer promising possibilities for such searches as they are sensitive to sources of CP violation that neutral kaon decays are not [6]. A signal of CP violation can be a difference in decay distributions between the charge conjugated decay modes. The main decay modes of the ground state hyperons are weak transitions into a baryon and a pseudoscalar meson like Λ → pπ − , branching fraction B ≈ 64 %, and Ξ − → Λπ − , B ≈ 100 % [7]. They involve two amplitudes: parity conserving to the relative p state, and parity violating to the s state. The angular distribution and the polarization of the daughter baryon are described by two decay parameters: the decay asymmetry α = 2Re(s * p)/(|p| 2 + |s| 2 ) and the relative phase φ = arg(s/p). Here, we denote decay asymmetries for Λ → pπ − and Ξ − → Λπ − as α Λ and α Ξ , respectively. In the CP symmetry conserving limit the parameters α and φ for the charge conjugated decay mode have the same absolute values but opposite signs e.g. α Λ = −αΛ. The best limit for CP violation in the strange baryon sector was obtained by comparing the Ξ − andΞ + decay chains of unpolarized Ξ baryons at the HyperCP (E871) experiment [8] by determining the asymmetry A ΞΛ = (α Λ α Ξ − αΛαΞ)/(α Λ α Ξ + αΛαΞ). The result, A ΞΛ = (0.0 ± 5.1 ± 4.7) × 10 −4 , is consistent with the SM predictions: |A ΞΛ | ≤ 5 × 10 −5 [9]. However, a preliminary HyperCP result presented at the BEACH 2008 Conference suggests a large value of the asymmetry A ΞΛ = (−6.0 ± 2.1 ± 2.0) × 10 −4 [10].With a well-defined initial state charmonium decay into a strange baryon-antibaryon pair offers an ideal system to test fundamental symmetries. Vector charmonia J/ψ and ψ can be directly produced in an elect...