High-order discretizations provide exceptional accuracy, but their computational efficiency depends critically on solvers that exploit local structure while controlling the cost of global coupling. Substructuring and hybridization address this challenge by organizing the computation around local problems coupled through interfaces, traces, or skeleton variables. Depending on the formulation, these interface unknowns arise from a decomposition of the discrete problem or are introduced as a part of the discretization itself. This minisymposium brings together research on the analysis, design, and implementation of solvers based on this local-to-global structure. It encompasses methods that eliminate or solve for interior unknowns locally and formulate the remaining global interaction as an interface problem. The emphasis is on the resulting solver architecture, whether the interface formulation is obtained algebraically, constructed through hybridization, or developed directly at the operator level. The minisymposium focuses on questions that become particularly important in the high-order regime: the construction and representation of interface operators, robustness with respect to discretization parameters and problem complexity, computational and memory efficiency, parallel scalability, and the interaction between local discretization structure and global solution strategies. Its aim is to connect researchers working with different formulations through the shared mathematical and computational principles of local elimination and interface coupling.