  In this article, we produce Grothendieck-Riemann-Roch formulas for cohomology
theories that are not oriented in the classical sense. We then specialize to
the case of cohomology theories that admit a so-called symplectic orientation
and show how to compute the relevant Todd classes in that situation. At the end
of the article, we illustrate our methods on the Borel character linking
Hermitian K-theory and rational MW-motivic cohomology.
