Here are the collection of energy levels produced by solving the 2-d oscillator in the r- coordinate system:
The wavefunctions for these levels look different: the x-y wavefunctions have square symmetry while the r- wavefunctions have circle-symmetry. Nevertheless, notice that for any value of E' the number of levels ("degeneracy") is the same in either system. For example, E'=10 includes 5 |nrm> wavefunctions: |0,-4>, |1,-2>, |2,0>, |1,2>, and |0,4> or includes 5 |nxny> wavefunctions: |0,4>, |1,3>, |2,2>, |3,1>, and |4,0> . In fact the set of functions spanned by forming all possible linear combinations of the 5 |nrm> wavefunctions is identical to that formed by all possible linear combinations of the 5 |nxny>. In particular any |nrm> wavefunction can be expressed as a linear combination of the co-degenerate |nxny> wavefunctions (and visa-versa). The degenerate x-y wavefunctions and the r- wavefunctions are thus different basis vectors for the same subspace.
Here is a start on finding the sum of |nxny> which equals any |nrm> for E'=10.