On the algebraic side, we obtain an Atiyah-Bott-Shapiro type isomorphism relating quaternionic modules over the Clifford algebras to symplectic K-theory of a point. On the topological side, we define a natural transformation from spin^h cobordism theory to symplectic K-theory, which in particular assigns to each spin^h manifold an integer or mod 2 valued cobordism invariant. These invariants can be expressed in terms of differential geometrical data using a quaternionic version of the index theorem.