  We consider BSDEs with two reflecting irregular barriers. We give necessary
and sufficient conditions for existence and uniqueness of $\mathbb{L}^{p}$
solutions for equations with generators monotone with respect to $y$ and
Lipschitz continuous with respect to $z$, and with data in $\mathbb{L}^{p}$
spaces for $p\ge 1$. We also prove that the solutions can be approximated via
penalization method.
