ABSTRACT 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 (DM) 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 coplanar orbits that are also perpendicular to the central stellar disc. We find that the normal to the common orbital plane of the coplanar satellites is well aligned with the minor axis of the host DM halo, with a median misalignment angle of only 17.3°. Based on this result, we infer that the minor axis of the Galactic DM halo points towards (l, b) = (182°, −2°), with an angular uncertainty at the 68 and 95 percentile confidence levels of 22° and 43°, 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°. This twist occurs over a very narrow radial range and reflects variations in the filamentary network along which mass was accreted into the MW.