The determination of the magnitude and "sign" of the J/psi polarization crucially depends on the reference frame used in the analysis of the data and a full understanding of the polarization phenomenon requires measurements reported in two "orthogonal" frames, such as the Collins-Soper and helicity frames. Moreover, the azimuthal anisotropy can be, in certain frames, as significant as the polar one. The seemingly contradictory J/psi polarization results reported by E866, HERA-B and CDF can be consistently described assuming that the most suitable axis for the measurement is along the direction of the relative motion of the colliding partons, and that directly produced J/psi's are longitudinally polarized at low momentum and transversely polarized at high momentum. We make specific predictions that can be tested on existing CDF data and by LHC measurements, which should show a full transverse polarization for direct J/psi mesons of pT > 25 GeV/c.