  We investigate the alignment of haloes with the filaments of the cosmic web
using an unprecedently large sample of dark matter haloes taken from the
P-Millennium $\Lambda$CDM cosmological N-body simulation. We use the
state-of-the-art NEXUS morphological formalism which, due to its multiscale
nature, simultaneously identifies structures at all scales. We find strong and
highly significant alignments, with both the major axis of haloes and their
peculiar velocity tending to orient along the filament. However, the spin -
filament alignment displays a more complex trend changing from preferentially
parallel at low masses to preferentially perpendicular at high masses. This
"spin flip" occurs at an average mass of $5\times10^{11}~h^{-1}M_\odot$. This
mass increases with increasing filament diameter, varying by more than an order
of magnitude between the thinnest and thickest filament samples. We also find
that the inner parts of haloes have a spin flip mass that is several times
smaller than that of the halo as a whole. These results confirm that recent
accretion is responsible for the complex behaviour of the halo spin - filament
alignment. Low-mass haloes mainly accrete mass along directions perpendicular
to their host filament and thus their spins tend to be oriented along the
filaments. In contrast, high-mass haloes mainly accrete along their host
filaments and have their spins preferentially perpendicular to them.
Furthermore, haloes located in thinner filaments are more likely to accrete
along their host filaments than haloes of the same mass located in thicker
filaments.
