We measure the dipion mass spectrum in X(3872)→J/ψπ+π− decays using 360  pb−1 of pp collisions at s√=1.96  TeV collected with the CDF II detector. The spectrum is fit with predictions for odd C-parity (S13, P11, and DJ3) charmonia decaying to J/ψπ+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 D0D*0 molecule. Only the S13 and J/ψρ hypotheses are compatible with our data. Since S13 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.