This work is intended to be a mathematical underpinning for the field of grain-boundary engineering and its relatives. The inter-relationships within the set of rotations producing coincident site lattices in cubic crystals are examined in detail. Besides combining previously established but widely scattered results into a unified context, the present work details newly developed representations of the group structure in terms of strings of generators (based on quaternionic number theory, and including uniqueness proofs and rules for algebraic manipulation) as well as an easily visualized topological network model. Important results that were previously obscure or not universally understood (e.g. the Σ combination rule governing triple junctions) are clarified in these frameworks. The methods also facilitate several general observations, including the very different natures of twin-limited structures in two and three dimensions, the inadequacy of the Σ combination rule to determine valid quadruple nodes, and a curious link between allowable grain-boundary assignments and the four-color map theorem. This kind of understanding is essential to the generation of realistic statistical models of grain-boundary networks (particularly in twin-dominated systems) and is especially applicable to the field of grain-boundary engineering.