  The relation between symmetry reduction before and after quantization of a
field theory is discussed using a toy model: the axisymmetric Klein-Gordon
field. We consider three possible notions of symmetry at the quantum level:
invariance under the group action, and two notions derived from imposing
symmetry as a system of constraints a la Dirac, reformulated as a first class
system. One of the latter two turns out to be the most appropriate notion of
symmetry in the sense that it satisfies a number of physical criteria,
including the commutativity of quantization and symmetry reduction. Somewhat
surprisingly, the requirement of invariance under the symmetry group action is
not appropriate for this purpose. A generalization of the physically selected
notion of symmetry to loop quantum gravity is presented and briefly discussed.
