  Intrinsic alignments of galaxies are a significant astrophysical systematic
affecting cosmological constraints from weak gravitational lensing. Obtaining
numerical predictions from hydrodynamical simulations of expected survey
volumes is expensive, and a cheaper alternative relies on populating large dark
matter-only simulations with accurate models of alignments calibrated on
smaller hydrodynamical runs. This requires connecting the shapes and
orientations of galaxies to those of dark matter haloes and to the large-scale
structure. In this paper, we characterise galaxy-halo alignments in the
Horizon-AGN cosmological hydrodynamical simulation. We compare the shapes and
orientations of galaxies in the redshift range $0<z<3$ to those of their
embedding dark matter haloes, and to the matching haloes of a twin dark-matter
only run with identical initial conditions. We find that galaxy ellipticities
in general cannot be predicted directly from halo ellipticities. The mean
misalignment angle between the minor axis of a galaxy and its embedding halo is
a function of halo mass, with residuals arising from the dependence of
alignment on galaxy type, but not on environment. Haloes are much more strongly
aligned among themselves than galaxies, and they decrease their alignment
towards low redshift. Galaxy alignments compete with this effect, as galaxies
tend to increase their alignment with haloes towards low redshift. We discuss
the implications of these results for current halo models of intrinsic
alignments and suggest several avenues for improvement.
