Structure of the decay ¯ B 0 s → J / ψ π + π − is studied using data corresponding to 3   fb − 1 of integrated luminosity from p p collisions produced by the LHC and collected by the LHCb detector. Five interfering π + π − states are required to describe the decay: f 0 ( 980 ) , f 0 ( 1500 ) , f 0 ( 1790 ) , f 2 ( 1270 ) , and f ′ 2 ( 1525 ) . An alternative model including these states and a nonresonant J / ψ π + π − component also provides a good description of the data. Based on the different transversity components measured for the spin-2 intermediate states, the final state is found to be compatible with being entirely C P odd. The C P -even part is found to be < 2.3 % at a 95% confidence level. The f 0 ( 500 ) state is not observed, allowing a limit to be set on the absolute value of the mixing angle with the f 0 ( 980 ) of < 7. 7 ∘ at a 90% confidence level, consistent with a tetraquark interpretation of the f 0 ( 980 ) substructure.