  We derive the Hamiltonian for trilayer moir\'e systems with the Coulomb
interaction projected onto the bands near the charge neutrality point.
Motivated by the latest experimental results, we focus on the twisted symmetric
trilayer graphene (TSTG) with a mirror-symmetry with respect to the middle
layer. We provide a full symmetry analysis of the non-interacting Hamiltonian
with a perpendicular displacement field coupling the band structure made
otherwise of the twisted bilayer graphene (TBG) and the high velocity Dirac
fermions, and we identify a hidden non-local symmetry of the problem. In the
presence of this displacement field, we construct an approximate
single-particle model, akin to the tripod model for TBG, capturing the essence
of non-interacting TSTG. We also derive more quantitative perturbation schemes
for the low-energy physics of TSTG with displacement field, obtaining the
corresponding eigenstates. This allows us to obtain the Coulomb interaction
Hamiltonian projected in the active band TSTG wavefunctions and derive the full
many-body Hamiltonian of the system. We also provide an efficient
parameterization of the interacting Hamiltonian. Finally, we show that the
discrete symmetries at the single-particle level promote the $\mathrm{U} \left(
2 \right) \times \mathrm{U} \left( 2 \right)$ spin-valley symmetry to enlarged
symmetry groups of the interacting problem under different limits. The
interacting part of the Hamiltonian exhibits a large $\mathrm{U} \left( 4
\right) \times \mathrm{U} \left( 4 \right) \times \mathrm{U} \left( 4 \right)
\times \mathrm{U} \left( 4 \right)$ symmetry in the chiral limit. Moreover, by
identifying a new symmetry which we dub spatial many-body charge conjugation,
we show that the physics of TSTG is symmetric around charge neutrality.
