KSp-characteristic classes determine Spin${}^h$ cobordism

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.