  In the context of extended Teleparallel gravity theories with a 3+1
dimensions Gauss-Bonnet analog term, we address the possibility of these
theories reproducing several well-known cosmological bouncing scenarios in a
four-dimensional Friedmann-Lema\^itre-Robertson-Walker geometry. We shall study
which types of gravitational Lagrangians are capable of reconstructing bouncing
solutions provided by analytical expressions for symmetric, oscillatory,
superbounce, the matter bounce and singular bounce. Some of the Lagrangians
discovered are both analytical at the origin having Minkowski and Schwarzschild
as vacuum solutions. All these results open the possibility up for such
theories to be competitive candidates of extended theories of gravity in
cosmological scales.
