This paper develops the category NCG. Its objects are node-and-choice games, which include essentially all extensive-form games. Its morphisms allow arbitrary transformations of a game's nodes, choices, and players, as well as monotonic transformations of the utility functions of the game's players. Among the morphisms are subgame inclusions. Several characterizations and numerous properties of the isomorphisms are derived. Also, the game-theoretic concepts of no-absentmindedness, perfect-information, and (pure-strategy) Nash-equilibrium are shown to be isomorphically invariant. Finally, full subcategories are defined for choice-sequence games and choice-set games, and relationships among these two subcategories and NCG itself are expressed and derived via isomorphic inclusions and equivalences.