  The resonant structure of the decay $\overline{B}_s^0\to J/\psi\pi^+\pi^-$ is
studied using data corresponding to 3 fb$^{-1}$ of integrated luminosity from
$pp$ collisions by the LHC and collected by the LHCb detector. Five interfering
$\pi^+\pi^-$ states are required to describe the decay:
$f_0(980),f_0(1500),f_0(1790),f_2(1270)$, and $f_2^{\prime}(1525)$. An
alternative model including these states and a non-resonant $J/\psi \pi^+\pi^-$
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 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 $<7.7^{\circ}$ at 90% confidence level, consistent
with a tetraquark interpretation of the $f_0(980)$ substructure.
