  We determine the spectra of a class of quantum spin chains of Temperley-Lieb
type by utilizing the concept of Temperley-Lieb equivalence with the S=1/2 XXZ
chain as a reference system. We consider open boundary conditions and in
particular periodic boundary conditions. For both types of boundaries the
identification with XXZ spectra is performed within isomorphic representations
of the underlying Temperley-Lieb algebra. For open boundaries the spectra of
these models differ from the spectrum of the associated XXZ chain only in the
multiplicities of the eigenvalues. The periodic case is rather different. Here
we show how the spectrum is obtained sector-wise from the spectra of globally
twisted XXZ chains. As a spin-off, we obtain a compact formula for the
degeneracy of the momentum operator eigenvalues. Our representation theoretical
results allow for the study of the thermodynamics by establishing a
TL-equivalence at finite temperature and finite field.
