By the 2040s, several all-sky surveys will have transformed our view of the large-scale structure. However, one of the major outstanding questions in astrophysics will remain: understanding how galaxies acquire and evolve their angular momentum and how this connects to the cosmic web. Measuring the alignments between galaxy spins and cosmic filaments across cosmic time, and understanding what this reveals about galaxy evolution, requires surveys that also characterise intrinsic alignments, i.e. correlations in galaxy shapes produced by the cosmic web itself rather than by lensing. Intrinsic alignments are a major source of systematic error in weak-lensing measurements of the fundamental parameters of the Universe. Addressing both questions together will necessitate new types of MOS surveys that combine kinematic information with high-completeness redshifts down to at least 24-25mag. To achieve our science goals, we require a new generation of wide-field spectroscopic facilities that can obtain spin-filament alignment measurements for millions of galaxies while simultaneously delivering sub-Mpc resolution of the cosmic web and spatially-resolved kinematics required to map the spin-filament connection at the level of individual galaxies within their local cosmic environment. Such a program would provide a unique legacy survey of galaxies and cosmic structures from kiloparsec to megaparsec scales, establishing ESO's leadership in bridging the physics of galaxy evolution with the systematic-control requirements for Stage-IV cosmological surveys.
