We study various forms of diagonal tetrads that accommodate Black Hole
solutions in $f(T)$ gravity with certain symmetries. As is well-known, vacuum
spherically symmetric diagonal tetrads lead to rather boring cases of constant
torsion scalars. We extend this statement to other possible horizon topologies,
namely, spherical, hyperbolic and planar horizons. All such cases are forced to
have constant torsion scalars to satisfy the anti-symmetric part of the field
equations. We give a full classification of possible vacuum static solutions of
this sort. Furthermore, we discuss addition of time-dependence in all the above
cases. We also show that if all the components of a diagonal tetrad depend only
on one coordinate, then the anti-symmetric part of the field equations is
automatically satisfied. This result applies to the flat horizon case with
Cartesian coordinates. For solutions with a planar symmetry (or a flat
horizon), one can naturally use Cartesian coordinates on the horizon. In this
case, we show that the presence of matter is required for existence of
non-trivial solutions. This is a novel and very interesting feature of these
constructions. We present two new exact solutions, the first is a magnetic
Black Hole which is the magnetic dual of a known electrically charged Black
Hole in literature. The second is a dyonic Black Hole with electric and
magnetic charges. We present some features of these Black holes, namely,
extremality conditions, mass, behavior of torsion and curvature scalars near
the singularity.
