The resonant structure of the reaction (B) over bar (0) --> J/psi pi(+)pi(-) is studied using data from 3 fb(-1) of integrated luminosity collected by the LHCb experiment, one third at 7 TeV center-of-mass energy and the remainder at 8 TeV. The invariant mass of the pi(+)pi(-) pair and three decay angular distributions are used to determine the fractions of the resonant and nonresonant components. Six interfering pi(+)pi(-) states, rho(770), f(0)(500), f(2)(1270), rho(1450), omega(782) and rho(1700), are required to give a good description of invariant mass spectra and decay angular distributions. The positive and negative charge parity fractions of each of the resonant final states are determined. The f(0)(980) meson is not seen and the upper limit on its presence, compared with the observed f(0)(500) rate, is inconsistent with a model where these scalar mesons are formed from two quarks and two antiquarks (tetraquarks) at the eight standard deviation level. In the q (q) over bar model, the absolute value of the mixing angle between the f(0)(980) and the f(0)(500) scalar mesons is limited to be less than 17 degrees at 90% confidence level.