We measure the dipion mass spectrum in X(3872)--> J/psipi(+) pi(-) decays using 360 pb(-1) of pp collisions at square root s= 1.96 TeV collected with the CDF II detector. The spectrum is fit with predictions for odd C-parity ((3)S(1), (1)P(1), and (3)D(J)) charmonia decaying to J/psipi(+) pi(-), as well as even C-parity states in which the pions are from rho(0) decay. The latter case also encompasses exotic interpretations, such as a D(0)D(*0) molecule. Only the (3)S(1) and J/psirho hypotheses are compatible with our data. Since (3)S(1) is untenable on other grounds, decay via J/psirho is favored, which implies C= +1 for the X(3872). Models for J/psi - rho different angular momenta L are considered. Flexibility in the models, especially the introduction of rho - omega interference, enables good descriptions of our data for both L = 0 and 1.
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