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Abstract: We study azimuthal asymmetries in diffractive J/ψ production in UPCs using CGC effective theory. We predict very large cos 2φ and cos 4φ azimuthal asymmetries for diffractive J/psi production both in UPCs and in eA collisions . These novel polarization dependent observables may provide complementary information for constraining gluon transverse spatial distribution inside large nuclei. As compared to all previous analysis of diffractive J/psi production, the essential new elements integrated in our theoretical calculations are: the double-slit interference effect, the linear polarization of coherent photons, and the final state soft photon radiation effect.