For G a classical group of type Bn, Cn or Dn defined over an algebraically closed field (of characteristic other than two if G is a special orthogonal group), we show that the centralizer algebra of G on the space of endomorphisms of the module of traceless tensors Wr is naturally isomorphic to the group algebra of the symmetric group Sr if r[les ]n (for G of type Bn or Cn) or if r<n (for G of type Dn). This extends a characteristic zero result of Brauer and Weyl and recovers some of the related characteristic free results that De Concini and Strickland obtained for G of type Cn.