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We characterise the cabling operations on the weight systems of finite type knot invariants. The eigenvectors and eigenvalues of this family of operations are described. The canonical deframing projection for these knot invariants is described over the cable eigenbasis. The action of immanent weight systems on general Feynman diagrams is considered, and the highest eigenvalue cabling eigenvectors are shown to be dual to the immanent weight systems. Using these results, we prove a recent conjecture of Bar-Natan and Garoufalidis on cablings of weight systems.