  Recent general results on Hamiltonian reductions under polar group actions
are applied to study some reductions of the free particle governed by the
Laplace-Beltrami operator of a compact, connected, simple Lie group. The
reduced systems associated with arbitrary finite dimensional irreducible
representations of the group by using the symmetry induced by twisted
conjugations are described in detail. These systems generically yield
integrable Sutherland type many-body models with spin, which are called twisted
spin Sutherland models if the underlying twisted conjugations are built on
non-trivial Dynkin diagram automorphisms. The spectra of these models can be
calculated, in principle, by solving certain Clebsch-Gordan problems, and the
result is presented for the models associated with the symmetric tensorial
powers of the defining representation of SU(N).
