From what I understand, the symmetry number for a molecule can be defined in 2 ways:
1. The quantum mechanical symmetry number corrects for overcounting the number of possible rotational states of a rotating molecule. Here's a source using this method.
2. The classical symmetry number corrects for including physical orientations of that molecule that appear the same because of its symmetrical physical structure. Here's a source using this method (a PDF file)
This source claims at the end of page 3, that the classical view is roughly based on the quantum view. I have a very hard time to grasp that.
How does the physical structure of a molecule have anything to do with nuclear spins being integer or half-integers, whether the total wavefunction needs to be symmetrical or antisymmetrical, and thus determining which rotational quantum states are allowed for a molecule?