Stephen Wolfram’s classification of elementary cellular automata into 88 equivalence classes marked a significant milestone in reducing and systematizing the study of one-dimensional cellular automaton rules. Motivated by this, we present a classification framework for two-dimensional complement-invariant binary cellular automata defined on a finite triangular lattice with a von Neumann neighborhood of radius one. The complete rule space consists of 65,536 rules. By introducing the concept of complement invariance, we consider 256 rules. Within the class of complement-invariant rules, each rule and its bitwise complement produce mutually complementary evolution patterns, which reduces the 256 rules to 128 equivalence classes under periodic boundary. Applying vertical symmetry refines the classification to a final set of 80 equivalence classes. Vertical symmetry works under both null and periodic boundaries. Our results demonstrate that complement symmetry is unique to complement-invariant rules and provide a systematic organization of complement-invariant CA on the triangular lattice. This work proves the isomorphism of vertically symmetric rules and identifies complement-invariant rules that generate self-similar fractal patterns.