  Trilayer graphene with a twisted middle layer has recently emerged as a new
platform exhibiting correlated phases and superconductivity near its magic
angle. A detailed characterization of its electronic structure in the parameter
space of twist angle $\theta$, interlayer potential difference $\Delta$, and
top-bottom layer stacking $\tau$ reveals that flat bands with large Coulomb
energy vs bandwidth $U/W > 1$ are expected within a range of $\pm 0.2^{\circ}$
near $\theta \simeq1.5^{\circ}$ and $\theta \simeq1.2^{\circ}$ for $\tau_{\rm
AA}$ top-bottom layer stacking, between a wider $1^{\circ} \sim 1.7^{\circ}$
range for $\tau_{\rm AB}$ stacking, whose bands often have finite valley Chern
numbers thanks to the opening of primary and secondary band gaps in the
presence of a finite $\Delta$, and below $\theta \lesssim 0.6^{\circ}$ for all
$\tau$ considered. The largest $U/W$ ratios are expected at the magic angle
$\sim 1.5^{\circ}$ when $|\Delta| \sim 0$~meV for AA, and slightly below near
$\sim 1.4^{\circ}$ for finite $|\Delta| \sim 25$~meV for AB stackings, and near
$\theta \sim 0.4^{\circ}$ for both stackings. When ${\tau}$ is the saddle point
stacking vector between AB and BA we observe pronounced anisotropic local
density of states (LDOS) strip patterns with broken triangular rotational
symmetry. We present optical conductivity calculations that reflect the changes
in the electronic structure introduced by the stacking and gate tunable system
parameters.
