  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 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 Gaussian random fields 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.
