  We present a tight-binding calculation of a twisted bilayer graphene at magic
angle $\theta\sim 1.08^\circ$, allowing for full, in- and out-of-plane,
relaxation of the atomic positions. The resulting band structure displays as
usual four narrow mini bands around the neutrality point, well separated from
all other bands after the lattice relaxation. A thorough analysis of the
mini-bands Bloch functions reveals an emergent $D_6$ symmetry, despite the lack
of any manifest point group symmetry in the relaxed lattice. The Bloch
functions at the $\Gamma$ point are degenerate in pairs, reflecting the
so-called valley degeneracy. Moreover, each of them is invariant under
C$_{3z}$, i.e., transforming like one-dimensional, in-plane symmetric
irreducible representation of an "emergent" $D_6$ group. Out of plane, the
lower doublet is even under C$_{2x}$, while the upper doublet is odd, which
implies that at least eight Wannier orbitals, two $s$-like and two $p_z$-like
for each of the two supercell sublattices AB and BA are necessary, probably not
sufficient, to describe the four mini bands. This unexpected one-electron
complexity is likely to play an important role in the still unexplained
metal-insulator-superconductor phenomenology of this system.
