We present an analysis of angular distributions and correlations of the X(3872) particle in the exclusive decay mode X(3872)-->J/psipi+ pi- with J/psi-->mu+ mu-. We use 780 pb-1 of data from pp[over ] collisions at sqrt[s]=1.96 TeV collected with the CDF II detector at the Fermilab Tevatron. We derive constraints on spin, parity, and charge conjugation parity of the X(3872) particle by comparing measured angular distributions of the decay products with predictions for different J(PC) hypotheses. The assignments J(PC)=1++ and 2-+ are the only ones consistent with the data.