We can mix states with different $m$ in hydrogen atom by rotating it around some axis (not coinciding with $z$). Thus rotation is the symmetry operation which mixes states with different $m$.
As hydrogen has hidden symmetry, which leads to degeneracy in $\ell$, there must be some symmetry operation, which would mix different $\ell$'s. What is this operation? I understand that this symmetry is related to Laplace-Runge-Lenz vector conservation, but I don't quite see what symmetry operation corresponds to it.