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.
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