  By applying a self-consistent Hartree-Fock approximation, we show that the
mechanism of dynamical symmetry breaking can account for the insulating phase
that develops about the charge neutrality point of twisted bilayer graphene
around the magic angle. ($i$) If the Coulomb interaction is screened by
metallic gates, the opening of a gap between the lowest-energy valence and
conduction bands proceeds through the breakdown of chiral symmetry at strong
coupling. Increasing the dielectric screening, however, we find a critical
coupling at which chiral symmetry breaking is suppressed, triggering a very
strong signal for time-reversal symmetry breaking with Haldane mass. ($ii$) If
the long-range tail of the Coulomb interaction is not screened, we see the
appearance of yet a different dominant pattern at strong coupling, which is
characterized by breaking the time-reversal invariance but with opposite flux
in the two sublattices of each carbon layer, with the consequent valley
symmetry breaking. In this case a gap is also opened in the Dirac cones, but
superposed to the splitting of the degeneracy of the low-energy bands at the
$K$ points of the moir\'e Brillouin zone.
