  Lattice dynamics play a crucial role in the physics of Moir\'e systems. In
twisted bilayer graphene (TBG), it was shown that, in addition to the graphene
phonons, there is another set of gapless excitations termed Moir\'e Phonons
[Phys. Rev. B, 075416, 2019] reflecting the lattice dynamics at the Moire
superlattice level. These modes were later suggested to be phasons due to the
incommensurate stacking of the two graphene layers [Phys. Rev. B, 155426,
2019]. In this work, we elucidate the equivalence of these two seemingly
distinct perspectives by identifying an underlying symmetry, which we dub
mismatch symmetry, that exists for any twist angle. For commensurate angles,
this is a discrete symmetry whereas for incommensurate angles, it is equivalent
to a continuous phase symmetry giving rise to phason modes. In the small angle
limit, such symmetry becomes a continuous local symmetry whose spontaneous
breaking gives rise to Moir\'e phonons as its Goldstone mode. We derive an
effective field theory for these collective modes in TBG in precise agreement
with the full model and discuss their different properties. Our analysis is
then generalized to twisted multilayer graphene (TMG) where we identify higher
order mismatch and deduce the count of gapless modes including graphene
phonons, Moir\'e phonons and phasons. Especially, we study twisted
mirror-symmetric trilayer graphene with an alternating twist angle $\theta$ and
find 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 mirror
sector with an additional set of gapped modes in the odd sector. Our work
presents a systematic study of lattice symmetries in TMG providing insights
into its unique lattice dynamics.
