  We investigate the realization of two bouncing paradigms, namely of the
superbounce and the loop quantum cosmological ekpyrosis, in the framework of
various modified gravities. In particular, we focus on the $F(R)$, $F(G)$ and
$F(T)$ gravities, and we reconstruct their specific subclasses which lead to
such universe evolutions. These subclasses constitute from power laws,
polynomials, or hypergeometric ansatzes, which can be approximated by power
laws. The qualitative similarity of different effective gravities which realize
the above two bouncing cosmologies, indicates to some universality lying behind
such a bounce. Finally, performing a linear perturbation analysis, we show that
the obtained solutions are conditionally or fully stable.
