Invariants of real vector bundles

For a compact smooth manifold with corners (or finite CW-complex) $X$, we can prescribe a finite set of spin or spin${}^h$ manifolds (possibly with boundary) mapping into it so that every real vector bundle over $X$ is determined, up to stable equivalence, by the Dirac indices of the real vector bundle when pulled-back onto those prescribed spin or spin${}^h$ manifolds.