We measure the dipion mass spectrum in X(3872) -> J/psi pi(+)pi(-) decays using 360 pb(-1) of (p) over barp collisions at 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/psi pi(+)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) over bar*(0) molecule. Only the (3)S(1) and J/psi rho hypotheses are compatible with our data. Since (3)S(1) is untenable on other grounds, decay via J/psi rho is favored, which implies C = +1 for the X(3872). Models for different J/psi-rho 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.