  We investigate the relationship between complex and symplectic cobordism
localized away from the prime~$2$ and show that these theories are related much
as a real Lie group is related to its complexification. This suggests that
ideas from the theory of symmetric spaces might be used to illuminate these
subjects. In particular, we give an explicit equivalence of ring spectra \[
MSp[1/2]\wedge Sp/U_+\simeq MU[1/2] \] and deduce that $MU[1/2]$ is a wedge of
copies of $MSp[1/2]$. We discuss the implications for the structure of the
stable operation algebra $MSp[1/2]^*MSp[1/2]$ and the dual cooperation algebra
$MSp[1/2]_*MSp[1/2]$. Finally we describe some related Witt vector algebra and
apply our results to the study of formal involutions on the category of formal
group laws over a $\mathbb{Z}[1/2]$-algebra.
