We investigate the interplay between the high- and low-energy phenomenology of CP-violating interactions of the Higgs boson with gauge bosons. For this purpose, we use an effective field theory approach and consider all dimension-six operators arising in so-called universal theories. We compute their loop-induced contributions to electric dipole moments and the CP asymmetry in B→X_{s}γ and compare the resulting current and prospective constraints to the projected sensitivity of the LHC. Low-energy measurements are shown to generally have a far stronger constraining power, which results in highly correlated allowed regions in coupling space-a distinctive pattern that could be probed at the high-luminosity LHC.