  We report on a fully self-consistent Hartree-Fock calculation of interaction
effects on the Moir\'e flat bands of twisted bilayer graphene, assuming that
valley U(1) symmetry is respected. We use realistic band structures and
interactions and focus on the charge neutrality point, where experiments have
variously reported either insulating or semimetallic behavior. Restricting the
search to orders for which the valley U(1) symmetry remains unbroken, we find
three types of self-consistent solutions with competitive ground state energy
(i) insulators that break $C_2 {\mathcal T}$ symmetry, including valley Chern
insulators (ii) spin or valley polarized insulators and (iii) rotation $C_3$
symmetry breaking semimetals whose gaplessness is protected by the topology of
the Moir\'e flat bands. We find that the relative stability of these states can
be tuned by weak strains that break $C_3$ rotation. The nematic semimetal and
also, somewhat unexpectedly, the $C_2 {\mathcal T}$ breaking insulators, are
stabilized by weak strain. These ground states may be related to the
semi-metallic and insulating behaviors seen at charge neutrality, and the
sample variability of their observation. We also compare with the results of
STM measurements near charge neutrality.
