  The spatial distribution of the satellite populations of the Milky Way and
Andromeda are puzzling in that they are nearly perpendicular to the disks of
their central galaxies. To understand the origin of such configurations we
study the alignment of the central galaxy, satellite system and dark matter
halo in the largest of the "Evolution and Assembly of GaLaxies and their
Environments" (EAGLE) simulation. We find that centrals and their satellite
systems tend to be well aligned with their haloes, with a median misalignment
angle of $33^{\circ}$ in both cases. While the centrals are better aligned with
the inner $10$ kpc halo, the satellite systems are better aligned with the
entire halo indicating that satellites preferentially trace the outer halo. The
central - satellite alignment is weak (median misalignment angle of
$52^{\circ}$) and we find that around $20\%$ of systems have a misalignment
angle larger than $78^{\circ}$, which is the value for the Milky Way. The
central - satellite alignment is a consequence of the tendency of both
components to align with the dark matter halo. As a consequence, when the
central is parallel to the satellite system, it also tends to be parallel to
the halo. In contrast, if the central is perpendicular to the satellite system,
as in the case of the Milky Way and Andromeda, then the central - halo
alignment is much weaker. Dispersion-dominated (spheroidal) centrals have a
stronger alignment with both their halo and their satellites than
rotation-dominated (disk) centrals. We also found that the halo, the central
galaxy and the satellite system tend to be aligned with the surrounding
large-scale distribution of matter, with the halo being the better aligned of
the three.
