Defect-free, charge-neutral, pure-carbon materials are generally viewed as intrinsically non-magnetic. Here we challenge this view by establishing a fundamental principle of symmetry-induced magnetism in pure-carbon fullerene systems through molecular orbital theory. We show that high-order symmetry of fullerene molecules or their crystalline lattices induces degenerate energy levels and hence, quantised magnetic moments at half filling. This mechanism holds for all high-order symmetries. It suggests a plausible intrinsic origin for the previously debated magnetic fullerene. Additionally, recent experimental advances in the synthesis of monolayer fullerene networks provide a feasible platform to implement our prediction on the deep link between symmetry and magnetism in these systems. Our results create new, broad frontiers of quantum magnetism by introducing molecular or crystalline symmetries in any lattice.