The J/ψ, ψ(3686)→Σ^{0}Σ[over ¯]^{0} processes and subsequent decays are studied using the world's largest J/ψ and ψ(3686) data samples collected with the BESIII detector. The parity-violating decay parameters of the decays Σ^{0}→Λγ and Σ[over ¯]^{0}→Λ[over ¯]γ, α_{Σ^{0}}=-0.0017±0.0021±0.0018 and α[over ¯]_{Σ^{0}}=0.0021±0.0020±0.0022, are measured for the first time. The strong CP symmetry is tested in the decays of the Σ^{0} hyperons for the first time by measuring the asymmetry A_{CP}^{Σ}=α_{Σ^{0}}+α[over ¯]_{Σ^{0}}=(0.4±2.9±1.3)×10^{-3}. The weak CP test is performed in the subsequent decays of their daughter particles Λ and Λ[over ¯]. Also for the first time, the transverse polarizations of the Σ^{0} hyperons in J/ψ and ψ(3686) decays are observed with opposite directions, and the ratios between the S-wave and D-wave contributions of the J/ψ, ψ(3686)→Σ^{0}Σ[over ¯]^{0} decays are obtained. These results are crucial to understand the decay dynamics of the charmonium states and the production mechanism of the Σ^{0}-Σ[over ¯]^{0} pairs.