Nilpotent cones (in reductive Lie algebras) are the basic linear-algebraic gadget underlying Hodge-theoretic compactifications of moduli spaces. They reflect the possible monodromies of multiparameter degenerations of varieties and their attached variations of Hodge structure. In this talk, we describe results with C. Robles and G. Pearlstein which provide a combinatorial algorithm for classifying nilpotent cones up to a suitable equivalency. The finite set resulting from this algorithm must often be further refined to arrive at the final list of Hodge-theoretically possible ("polarizable") nilpotent cones. For Hodge structures of Calabi--Yau type, mirror symmetry gives one way to prove polarizability, and we will demonstrate this in a case which is closely related to local CY 3-folds and a family of Feynman integrals studied in recent work with S. Bloch and P. Vanhove.