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We measure the dipion mass spectrum in X(3872) → J/ψ π + π - decays using 360 pb-1 of p̅p collisions at √s = 1.96 TeV collected with the CDF II detector. The spectrum is fit with predictions for odd C-parity (3S1, 1P1, and 3DJ) charmonia decaying to J/ π + π -, as well as even C-parity states in which the pions are from ρ0 decay. The latter case also encompasses exotic interpretations, such as a D0D̅*0 molecule. Only the 3S1 and J/ψρ hypotheses are compatible with our data. Since 3S1 is untenable on other grounds, decay via J/ψρ is favored, which implies C= +1 for the X(3872). Models for different J/ψ-ρ angular momenta L are considered. Flexibility in the models, especially the introduction of ρ- ω interference, enables good descriptions of our data for both L = 0 and 1.