  The proximity-induced spin-orbit coupling (SOC) in heterostructures of
twisted graphene and topological insulators (TIs) Bi$_2$Se$_3$ and Bi$_2$Te$_3$
is investigated from first principles. To build commensurate supercells, we
strain graphene and correct thus resulting band offsets by applying a
transverse electric field. We then fit the low-energy electronic spectrum to an
effective Hamiltonian that comprises orbital and spin-orbit terms. For twist
angles 0$^\circ\leq\Theta \lessapprox 20^\circ$, we find the dominant
spin-orbit couplings to be of the valley-Zeeman and Rashba types, both a few
meV strong. We also observe a sign change in the induced valley-Zeeman SOC at
$\Theta\approx 10^\circ$. Additionally, the in-plane spin structure resulting
from the Rashba SOC acquires a non-zero radial component, except at $0^\circ$
or $30^\circ$. At $30^\circ$ the graphene Dirac cone interacts directly with
the TI surface state. We therefore explore this twist angle in more detail,
studying the effects of gating, TI thicknesses, and lateral shifts on the SOC
parameters. We find, in agreement with previous results, the emergence of the
proximitized Kane-Mele SOC, with a change in sign possible by electrically
tuning the Dirac cone within the TI bulk band gap.
