spin/ angular momentum, representations of SO(3), SU(2)

plmokn2

1. The problem statement, all variables and given/known data
I'm trying to understand why particles have both spin and angular momentum in terms of group theory. As I understand it orbital angular momentum comes from the normal generators SO(3) which are intuitively infintesimal rotations so d/d(theta) etc. Also spin comes from representations of SO(3) in terms of SU(2j+1) so generators of SU are equivilent to generators of SO(3) so are conserved.

The thing is I don't really get why a particle only has one spin degree of freedom (so particles are only ever spin 1/2 or spin 1 or whatever) which is equivilant to one of the SU representations but particles also have orbital angular momentum as well as spin.

3. The attempt at a solution
It seems reasonable, looking at the operators, that spin is a fundamentally different from orbital angular momentum, but I can't see why this should be the case.

Any help appreciated.
Thanks