  Twisted trilayer graphene is a particularly promising moir\'e superlattice
system, due to its tunability, strong superconductivity, and complex electronic
symmetry breaking. Motivated by these properties, we study lattice relaxation
and the long-wavelength phonon modes of this system. We show that
mirror-symmetric trilayer graphene hosts, aside from the conventional acoustic
phonon modes, two classes of shear modes, which are even and odd under mirror
reflection. The mirror-even modes are found to be gapless and equivalent to the
"phason" modes of twisted bilayer graphene, with appropriately rescaled
parameters. The modes odd under mirror symmetry have no analogue in twisted
bilayer graphene and exhibit a finite gap, which we show is directly
proportional to the degree of lattice relaxation. We also discuss the impact of
mirror-symmetry breaking, which can be tuned by a displacement field or result
from a stacking shift, and of rotational- as well as time-reversal-symmetry
breaking, resulting from spontaneous electronic order. We demonstrate that this
can induce finite angular momentum to the phonon branches. Our findings are
important to the interpretation of recent experiments, concerning the origin of
superconductivity and of linear-in-$T$ resistivity.
