I think I must recant my comment on spin. I was thinking of a spin-1/2 particle. Its state lives in a 2-dimensional Hilbert space. If you rotate your spatial coordinates, there is a corresponding transformation of the basis of the 2-dimensional Hilbert space. Any change of basis for this Hilbert space can be obtained in this way. However, for a spin-n particle, the Hilbert space is 2n+1 dimensional, and I think there are many bases one cannot transform into by the transformations that are induced by a spatial rotation. As a consequence, for spin-n with n > 1/2 I think there are some bases which are not eigenbases of any angular momentum operator, and so could be considered in some sense "not preferred."