  In teleparallel gravity, we apply Lorenz type gauge fixing to cope with
redundant degrees of freedom in the vierbein field. This condition is mainly to
restore the Lorentz symmetry of the teleparallel torsion scalar. In
cosmological application, this technique provides standard cosmology,
turnaround, bounce or $\Lambda$CDM as separate scenarios. We reconstruct the
$f(T)$ gravity which generates these models. We study the stability of the
solutions by analyzing the corresponding phase portraits. Also, we investigate
Lorenz gauge in the unimodular coordinates, it leads to unify a nonsingular
bounce and standard model cosmology in a single model, where crossing the
phantom divide line is achievable through a finite-time singularity of Type IV
associated with a de Sitter fixed point. We reconstruct the unimodular $f(T)$
gravity which generates the unified cosmic evolution showing the role of the
torsion gravity to establish a healthy bounce scenario.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Theory