In our QM class, the prof said:
"We are ready to begin constructing the individual states of the 3D isotropic harmonic oscillator system. The key property is that the states must organize themselves into representations of angular momentum. Since angular momentum commutes with the Hamiltonian, angular momentum multiplets represent degenerate states."
Why do they "must"? Is that because there is rotational invariance and therefore there "must" be a conserved angular momentum, hence an algebra of angular momentum operators, which necessarily lead to $|\ell,m\rangle$ eigenkets spanning the space of states for the 3D oscillator?