We study the alignments of galaxy spin axes with respect to cosmic web filaments as a function of various properties of the galaxies and their constituent bulges and discs. We exploit the SAMI Galaxy Survey to identify 3D spin axes from spatially-resolved stellar kinematics and to decompose the galaxy into the kinematic bulge and disc components. The GAMA survey is used to reconstruct the cosmic filaments. The mass of the bulge, defined as the product of stellar mass and bulge-to-total flux ratio Mbulge = M⋆ × (B/T), is the primary parameter of correlation with spin–filament alignments: galaxies with lower bulge masses tend to have their spins parallel to the closest filament, while galaxies with higher bulge masses are more perpendicularly aligned. M⋆ and B/T separately show correlations, but they do not fully unravel spin–filament alignments. Other galaxy properties, such as visual morphology, stellar age, star formation activity, kinematic parameters and local environment, are secondary tracers. Focussing on S0 galaxies, we find preferentially perpendicular alignments, with the signal dominated by high-mass S0 galaxies. Studying bulge and disc spin–filament alignments separately reveals additional information about the formation pathways of the corresponding galaxies: bulges tend to have more perpendicular alignments, while discs show different tendencies according to their kinematic features and the mass of the associated bulge. The observed correlation between the flipping of spin–filament alignments and the growth of the bulge can be explained by mergers, which drive both alignment flips and bulge formation.