Large-angle twisted graphene lies beyond the local-stacking description of conventional moiré systems: inequivalent rotation centers define distinct commensurate interfaces whose low-energy interlayer hybridization is governed by intervalley Umklapp tunneling. For three-dimensional twisted graphite assembled from interfaces with different crystalline symmetries, a symmetry-constrained effective model reveals that competition between nonchiral and chiral tunneling produces one-ring and two-ring nodal-line phases and a C₃-protected higher-order topological insulator in the presence of sublattice (chiral) symmetry. The nodal rings carry integer winding numbers, allowing oppositely wound rings to annihilate into the gapped phase. We further examine how sublattice-symmetry breaking modifies these phases. Density functional theory (DFT) and atomistic calculations for 21.8^∘ twisted graphite identify the equilibrium structure as a higher-order topological insulator, while compression drives it into a topological Weyl semimetal phase. These findings establish the stacking sequence of symmetry-inequivalent interfaces as a means of engineering band topology in three-dimensional twisted structures.