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Lattice dynamics play a crucial role in the physics of Moir\'e systems. Intwisted bilayer graphene (TBG), it was shown that, in addition to the graphenephonons, there is another set of gapless excitations termed Moir\'e Phonons[Phys. Rev. B, 075416, 2019] reflecting the lattice dynamics at the Moiresuperlattice level. These modes were later suggested to be phasons due to theincommensurate stacking of the two graphene layers [Phys. Rev. B, 155426,2019]. In this work, we elucidate the equivalence of these two seeminglydistinct perspectives by identifying an underlying symmetry, which we dubmismatch symmetry, that exists for any twist angle. For commensurate angles,this is a discrete symmetry whereas for incommensurate angles, it is equivalentto a continuous phase symmetry giving rise to phason modes. In the small anglelimit, such symmetry becomes a continuous local symmetry whose spontaneousbreaking gives rise to Moir\'e phonons as its Goldstone mode. We derive aneffective field theory for these collective modes in TBG in precise agreementwith the full model and discuss their different properties. Our analysis isthen generalized to twisted multilayer graphene (TMG) where we identify higherorder mismatch and deduce the count of gapless modes including graphenephonons, Moir\'e phonons and phasons. Especially, we study twistedmirror-symmetric trilayer graphene with an alternating twist angle $\theta$ andfind that it can be mapped to a TBG with the re-scaled twist angle$\sqrt{2/3}\theta$, hosting the same Moir\'e phonon modes in the even mirrorsector with an additional set of gapped modes in the odd sector. Our workpresents a systematic study of lattice symmetries in TMG providing insightsinto its unique lattice dynamics.

Symmetry Origin of Lattice Vibration Modes in Twisted Multilayer Graphene: Phasons vs Moir\'e Phonons