  The proxy-SU(3) symmetry has been proposed for spin-orbit like nuclear shells
using the asymptotic deformed oscillator basis for the single particle
orbitals, in which the restoration of the symmetry of the harmonic oscillator
shells is achieved by a change of the number of quanta in the z-direction by
one unit for the intruder parity orbitals. The same definition suffices within
the cartesian basis of the Elliott SU(3) model. Through a mapping of the
cartesian Elliott basis onto the spherical shell model basis, we translate the
proxy-SU(3) approximation into spherical coordinates, proving, that in the
spherical shell model basis the proxy-SU(3) approximation corresponds to the
replacement of the intruder parity orbitals by their de Shalit--Goldhaber
partners. Furthermore it is shown, that the proxy-SU(3) approximation in the
cartesian Elliott basis is equivalent to a unitary transformation in the
z-coordinate, leaving the x-y plane intact, a result which in the asymptotic
deformed oscillator coordinates implies, that the z-projections of angular
momenta and spin remain unchanged. The present work offers a microscopic
justification of the proxy-SU(3) approximation and in addition paves the way,
for taking advantage of the proxy-SU(3) symmetry in shell model calculations.
