The twig edge states in graphene-like structures are viewed as the fourth states complementary to their zigzag, bearded, and armchair counterparts. In this work, we study a rod-in-plasma system in honeycomb lattice with twig edges under external magnetic fields and lattice scaling and show that twig edge states can exist in different phases of the system, such as quantum Hall phase, quantum spin Hall phase and insulating phase. The twig edge states in the quantum Hall phase exhibit robust one-way transmission property immune to backscattering and thus provide a novel avenue for solving the plasma communication blackout problem. Moreover, we demonstrate that corner and edge states can exist within the trivial band gap of the insulating phase by modulating the on-site potential of the twig edges. Especially, helical edge states with the unique feature of pseudospin-momentum locking that could be exited by chiral sources are demonstrated at the twig edges within the trivial band gap. Our results show that many topological-like behaviors of electromagnetic waves are not necessarily tied to the exact topology of the systems and the twig edges and interface engineering can bring new opportunities for more flexible manipulation of electromagnetic waves.