The nature of magnetism in layered Ruddlesden-Popper nickelates remains a central open question, particularly in light of recent observations of spin-wave-like magnetic excitations in metallic multilayer compounds. Here, we develop a unified itinerant description of spin-density-wave (SDW) order and magnetic excitations in La₃Ni₂O₇ and La₄Ni₃O₁₀. The essential ingredient is the multilayer mirror structure of the NiO₂ blocks, which organizes the low-energy electronic states into mirror-even and mirror-odd sectors. We show that dominant interband nesting between mirror-opposite bands drives a mirror-selective itinerant SDW instability, whose collective modes naturally reproduce the experimentally observed spin-wave-like spectra. In La₄Ni₃O₁₀, the SDW further induces a secondary mirror-even charge density wave, yielding intertwined spin and charge textures. Our results demonstrate that magnetism in multilayer nickelates is fundamentally itinerant rather than local-moment in origin, and establish mirror-selective interband SDW order as a unifying organizing principle for magnetic correlations in these systems.