Based on (2712.4±14.1)×106 ψ(3686) decays collected with the BESIII detector, we have observed, for the first time, the hadronic decays of S- and P-wave charmonium states into 2(π+π-)η via radiative transitions from ψ(3686). The branching fraction of the decay ηc(1S)→2(π+π-)η has a significant dependence on the interference pattern between ηc(1S) and non-ηc(1S) processes. We measure it in both the destructive and constructive interference scenarios for the first time. The mass and width of the ηc(1S) are measured to be M=(2984.14±0.13±0.38) MeV/c2 and Γ=(28.82±0.11±0.82) MeV, respectively. Clear signals for the decays of the χcJ(J=0,1,2) and the ηc(2S) to 2(π+π-)η are also observed for the first time, and the corresponding branching fractions are measured. The ratio of the branching fractions between the ηc(2S) and ηc(1S) decays is significantly lower than the theoretical prediction, which might suggest different dynamics in their decays.