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The close relationship between mergers and the reorientation of the spin for galaxies and their host dark haloes is investigated using a cosmological hydrodynamical simulation (Horizon-AGN). Through a statistical analysis of merger trees, we show that spin swings are mainly driven by mergers along the filamentary structure of the cosmic web, and that these events account for the preferred perpendicular orientation of massive galaxies with respect to their nearest filament. By contrast, low-mass galaxies (Ms < 1010 M⊙ at redshift 1.5) having undergone very few mergers, if at all, tend to possess a spin well aligned with their filament. Haloes follow the same trend as galaxies but display a greater sensitivity to smooth anisotropic accretion. The relative effect of mergers on magnitude is qualitatively different for minor and major mergers: mergers (and diffuse accretion) generally increase the magnitude of the specific angular momentum, but major mergers also give rise to a population of objects with less specific angular momentum left. Without mergers, secular accretion builds up the specific angular momentum of galaxies but not that of haloes. It also (re)aligns galaxies with their filament. [ABSTRACT FROM PUBLISHER]