  We show that a representable motivic cohomology theory admits a unique
normalized SL^c-orientation if the zeroth cohomology presheaf is a Zariski
sheaf. We also construct Thom isomorphisms in SL-oriented cohomology for
SL^c-bundles and obtain new results on the \eta-torsion characteristic classes,
in particular, we prove that the Euler class of an oriented bundle admitting a
(possibly non-orientable) odd rank subbundle is annihilated by the Hopf
element.
