Memristive devices have revolutionized non-volatile memory and neuromorphic computing, yet the geometry of their hysteresis loops, in particular, the occurrence and robustness of multiple selfcrossings, remains poorly understood. Here we introduce a geometric and algebraic framework in which the crossing number, the number of transverse self-intersections of a memristor hysteresis loop, serves as a robust integer-valued geometric index. Drawing on singularity theory and cusp catastrophe, we employ discriminants and resultants to stratify the six-dimensional parameter space. This approach partitions the parameter space into structurally stable regions separated by explicitly computable transition boundaries. We demonstrate that the crossing number remains invariant under continuous parameter variations within structurally stable regions and changes only when a bifurcation boundary associated with a self-tangency or cusp singularity is crossed, thereby providing a complete classification within the polynomial subclass considered here. These insights bridge device physics with modern singularity theory and suggest a clear roadmap for exploiting higher-order memory effects in next-generation electronics and brain-inspired hardware.