Powers of the Dirac Operator on $S^1\times S^{n-1}$

We give explicit formulas for the spectra of the intertwinors on the spinor bundle over $S^1 \times S^{n-1}$, $n$ even, with the standard Lorentzian metric $g=-g_{{}_{S^1}}+g_{{}_{S^{n-1}}}$. As a special case, we construct conformally covariant differential operators of all odd orders with the leading term a power of the Dirac operator.