The geometry of the cosmic web drives in part the spin acquisition of galaxies. This can be explained in a Lagrangian framework, by identifying the specific long-wavelength correlations within the primordial Gaussian random field (GRF), which are relevant to spin acquisition. Tidal torque theory is revisited in the context of such anisotropic environments, biased by the presence of a filament within a wall. The point process of filament-type saddles represents it most efficiently. The constrained misalignment between the tidal and the inertia tensors in the vicinity of filament-type saddles simply explains the distribution of spin directions. This misalignment implies in particular an azimuthal orientation for the spins of more massive galaxies and a spin alignment with the filament for less massive galaxies. This prediction is found to be in qualitative agreement with measurements in GRFs and N-body simulations. It relates the transition mass to the geometry of the saddle, and accordingly predicts its measured scaling with the mass of non-linearity. Implications for galaxy formation and weak lensing are briefly discussed, as is the dual theory of spin alignments in walls.