  Teleparallel Gravity offers the possibility of reformulating gravity in terms
of torsion by exchanging the Levi-Civita connection with the Weitzenb\"ock
connection which describes torsion rather than curvature. Surprisingly,
Teleparallel Gravity can be formulated to be equivalent to general relativity
for a appropriate setup. Our interest lies in exploring an extension of this
theory in which the Lagrangian takes the form of $f(T,B)$ where T and B are two
scalars that characterize the equivalency with general relativity. In this
work, we explore the possible of reproducing well-known cosmological bouncing
scenarios in the flat Friedmann-Lema\^itre-Robertson-Walker geometry using this
approach to gravity. We study the types of gravitational Lagrangians which are
capable of reconstructing analytical solutions for symmetric, oscillatory,
superbounce, matter bounce, and singular bounce settings. These new
cosmologically inspired models may have an effect on gravitational phenomena at
other cosmological scales.
