Structure of the decay B¯s0→J/ψπ+π- is studied using data corresponding to 3 fb−1 of integrated luminosity from pp collisions produced by the LHC and collected by the LHCb detector. Five interfering π+π- states are required to describe the decay: f0(980),f0(1500),f0(1790),f2(1270), and f2′(1525). An alternative model including these states and a nonresonant J/ψπ+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 CP odd. The CP-even part is found to be <2.3% at a 95% confidence level. The f0(500) state is not observed, allowing a limit to be set on the absolute value of the mixing angle with the f0(980) of <7.7° at a 90% confidence level, consistent with a tetraquark interpretation of the f0(980) substructure.