  The Schr\"odinger operators with exchange terms for certain
Calogero-Sutherland quantum many body systems have eigenfunctions which factor
into the symmetric ground state and a multivariable polynomial. The polynomial
can be chosen to have a prescribed symmetry (i.e. be symmetric or
antisymmetric) with respect to the interchange of some specified variables. For
four particular Calogero-Sutherland systems we construct an eigenoperator for
these polynomials which separates the eigenvalues and establishes
orthogonality. In two of the cases this involves identifying new operators
which commute with the corresponding Schr\"odinger operators. In each case we
express a particular class of the polynomials with prescribed symmetry in a
factored form involving the corresponding symmetric polynomials.
