  We investigate the interplay between the high- and low-energy phenomenology
of $C\!P$-violating interactions of the Higgs boson with gauge bosons. For this
purpose we use an effective field theory approach and consider all dimension-6
operators arising in so-called universal theories. We compute their
loop-induced contributions to electric dipole moments and the $C\!P$ asymmetry
in $B\to X_s\gamma$, 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.
