What symmetry does conserved $L^2$ imply?

By the derivative rule $$ [H,L^2]=[H,L]L+L[H,L] $$ so there is noting new in $[H,L^2]$ that is not in $[H,L]$.

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Edit: in fact my answer is not correct. Take $H=L_x$. Then $[L^2,L_x]=0$ but $[L,L_x]\ne 0$.

So: if $[L,H]=0$ then $[L^2,H]=0$ implies nothing more, but it could be that $[L^2,H]=0$ without $[L,H]=0$.