  We show that the $e$-$e$ interaction induces a strong breakdown of valley
symmetry for each spin channel in twisted trilayer graphene, leading to a
ground state where the two spin projections have opposite sign of the valley
symmetry breaking order parameter. This leads to a spin-valley locking in which
the electrons of a Cooper pair are forced to live on different Fermi lines
attached to opposite valleys. Furthermore, we find an effective intrinsic
spin-orbit coupling explaining the protection of the superconductivity against
in-plane magnetic fields. The effect of spin-selective valley symmetry breaking
is validated as it reproduces the experimental observation of the reset of the
Hall density at 2-hole doping. It also implies a breakdown of the symmetry of
the bands from $C_6$ to $C_3$, with an enhancement of the anisotropy of the
Fermi lines which is at the origin of a Kohn-Luttinger (pairing) instability.
The isotropy of the bands is gradually recovered, however, when the Fermi level
approaches the bottom of the second valence band, explaining why the
superconductivity fades away in the doping range beyond 3 holes per moir\'e
unit cell in twisted trilayer graphene.
