  We show that the superconducivity in twisted graphene multilayers originates
from a common feature, which is the strong valley symmetry breaking
characteristic of these moir\'e systems at the magic angle. This leads to a
breakdown of the rotational symmetry of the flat moir\'e bands down to $C_3$,
and to ground states in which the time-reversal symmetry is broken for a given
spin projection. However, this symmetry can be recovered upon exchange of
spin-up and spin-down electrons, as we illustrate by means of a self-consistent
microscopic Hartree-Fock resolution where the states for the two spin
projections acquire opposite sign of the valley polarization. There is then a
spin-valley locking by which the Fermi lines for the two spin projections are
different and related by inversion symmetry. This effect represents a large
renormalization of the bare spin-orbit coupling of the graphene multilayers,
lending protection to the superconductivity against in-plane magnetic fields.
In the twisted bilayer as well as in trilayer and quadrilayer graphene, the
pairing glue is shown to be given by the nesting between parallel segments of
the Fermi lines which arise from the breakdown of symmetry down to $C_3$. This
leads to a strong Kohn-Luttinger pairing instability, which is relevant until
the Fermi line recovers gradually a more isotropic shape towards the bottom of
the second valence band, explaining why the superconductivity fades away beyond
three-hole doping of the moir\'e unit cell.
