  In this paper we determine the spectral data parametrizing Higgs bundles in a
generic fiber of the Hitchin map for the case where the structure group is the
special Clifford group with fixed Clifford norm. These are spin and "twisted"
spin Higgs bundles. The method used relates variations in spectral data with
respect to the Hecke transformations for orthogonal bundles introduced by Abe.
The explicit description also recovers a result from the geometric Langlands
program which states that the fibers of the Hitchin map are the dual abelian
varieties to the corresponding fibers of the moduli spaces of projective
orthogonal Higgs bundles (in the even case) and projective symplectic Higgs
bundles (in the odd case).
