We measure the anisotropy of dark matter flows on small scales in the near environment of haloes using a large set of simulations. We rely on two different approaches to quantify the anisotropy of the cosmic infall: we measure the flows at the haloes' virial radius while describing the infalling matter via fluxes through a spherical shell; we measure the spatial and kinematical distributions of satellites and substructures around haloes detected by the subclump finder ADAPTAHOP first described in the appendix. The two methods are found to be in agreement both qualitatively and quantitatively via one and two points statistics.The infall takes place preferentially in the plane perpendicular to the direction defined by the halo's spin. We computed the excess of equatorial accretion both through rings and via a harmonic expansion of the infall. The level of anisotropy of infalling matter is found to be ~15 %. The substructures have their spin orthogonal to their velocity vector in the halo's rest frame at a level of about 5%, suggestive of an image of a flow along filamentary structures which provides an explanation for the measured anisotropy. We conclude that a halo does not see its environment as an isotropic perturbation, investigate how the anisotropy is propagated inwards using perturbation theory, and discuss briefly implications for weak lensing, warps and the thickness of galactic disks.