  This paper examines a number of related questions about Euler characteristics
and characteristic classes with values in Witt cohomology. We establish a
motivic version of the Becker-Gottllieb transfer, generalizing a construction
of Hoyois.
  Ananyevskiy's splitting principle reduces questions about characteristic
classes of vector bundles in $\text{SL}$-oriented, $\eta$-invertible theories
to the case of rank two bundles. We refine the torus-normalizer splitting
principle for $\text{SL}_2$ to help compute the characteristic classes in Witt
cohomology of symmetric powers of a rank two bundle, and then generalize this
to develop a general calculus of characteristic classes with values in Witt
cohomology.
