  The interlayer van der Waals interaction in twisted bilayer graphene (tBLG)
induces both in-plane and out-of-plane atomic displacements exhibiting complex
patterns that depend on the twist angle. In particular, for small twist angles,
within each graphene layer, the relaxations give rise to a vortex displacement
pattern which is known to affect the dispersion of the flat band. Here, we
focus on yet another structural property, the chirality of the twisted bilayer.
We perform first-principles calculations based on density functional theory,
starting from the ground state optimized structures of tBLG at different twist
angles. We study the interplay between twist chirality and the atomic
relaxation patterns. Furthermore, we investigated the spin textures around the
$K$ points of the Brillouin zone, showing that alternating vortex-like patterns
are correlated to the chirality of tBLG. Interestingly, the sense of rotation
of each vortex is inverted by changing the chirality in tBLG while the
different twist angles also modify the spin textures. We discuss the origin of
the spin textures by calculating the layer weights and using plot regression
models.
