  We study the inequivalent quantizations of the N = 3 Calogero model by
separation of variables, in which the model decomposes into the angular and the
radial parts. Our inequivalent quantizations respect the ` mirror-S_3\rq\
invariance (which realizes the symmetry under the cyclic permutations of the
particles) and the scale invariance in the limit of vanishing harmonic
potential. We find a two-parameter family of novel quantizations in the angular
part and classify the eigenstates in terms of the irreducible representations
of the S_3 group. The scale invariance restricts the quantization in the radial
part uniquely, except for the eigenstates coupled to the lowest two angular
levels for which two types of boundary conditions are allowed independently
from all upper levels. It is also found that the eigenvalues corresponding to
the singlet representations of the S_3 are universal (parameter-independent) in
the family, whereas those corresponding to the doublets of the S_3 are
dependent on one of the parameters. These properties are shown to be a
consequence of the spectral preserving SU(2) (or its subrgoup U(1))
transformations allowed in the family of inequivalent quantizations.
