  A classic result of Anderson, Brown, and Peterson states that the cobordism
spectrum MSpin (respectively, MSpin$^c$) splits as a sum of Eilenberg--Mac Lane
spectra and connective covers of real K-theory (respectively, complex K-theory)
at 2. We develop a theory of symplectic K-theory classes and use these to build
an explicit splitting for MSpin$^h$ in terms of Eilenberg--Mac Lane spectra and
spectra related to symplectic K-theory. This allows us to determine the
Spin$^h$ cobordism groups systematically. We also prove that two
Spin$^h$-manifolds are cobordant if and only if their underlying unoriented
manifolds are cobordant and their KSp-characteristic numbers agree.
