  We analyse systems analogous to the Milky Way (MW) in the EAGLE cosmological
hydrodynamics simulation in order to deduce the likely structure of the MW's
dark matter halo. We identify MW-mass haloes in the simulation whose satellite
galaxies have similar kinematics and spatial distribution to those of the
bright satellites of the MW, specifically systems in which the majority of the
satellites (8 out of 11) have nearly co-planar orbits that are also
perpendicular to the central stellar disc. We find that the normal to the
common orbital plane of the co-planar satellites is well aligned with the minor
axis of the host dark matter halo, with a median misalignment angle of only
$17.3^\circ$. Based on this result, we infer that the minor axis of the
Galactic dark matter halo points towards $(l,b)=(182^\circ,-2^\circ)$, with an
angular uncertainty at the 68 and 95 percentile confidence levels of 22$^\circ$
and 43$^\circ$ respectively. Thus, the inferred minor axis of the MW halo lies
in the plane of the stellar disc. The halo, however, is not homologous and its
flattening and orientation vary with radius. The inner parts of the halo are
rounder than the outer parts and well-aligned with the stellar disc (that is
the minor axis of the halo is perpendicular to the disc). Further out, the halo
twists and the minor axis changes direction by $90^\circ$. This twist occurs
over a very narrow radial range and reflects variations in the filamentary
network along which mass was accreted into the MW.
