Real graded Clifford-module bundles and their Grothendieck groups

Let $V\to X$ be a finite-rank Euclidean real vector bundle and put $C(V)=\mathrm{C}\mathrm{l}(V,-⟨\cdot ,\cdot ⟩)$, so each vector $v$ acts with square $-\parallel v{\parallel }^2$. An ungraded $C(V)$-module bundle is a finite-rank real vector bundle with continuous fiberwise unital Clifford action. Write $K_{C(V)}(X)$ for the Grothendieck group of their direct-sum monoid.

A graded $C(V)$-module bundle is a module $\mathrm{\Lambda }={\mathrm{\Lambda }}^0\oplus {\mathrm{\Lambda }}^1$ whose action satisfies $C(V{)}^i{\mathrm{\Lambda }}^j\subseteq {\mathrm{\Lambda }}^{i+j}$, with indices modulo two. In particular vector multiplication is odd: it exchanges the two parity subbundles. Write $M(V)$ for the Grothendieck group of graded modules. The same convention gives $M_n$ for the negative Clifford algebra $C_n={\mathrm{C}\mathrm{l}}_{0,n}$ over a point. Ungraded $K_{C(V)}(X)$ and graded $M(V)$ are distinct groups; neither is identified with the other by notation. The rank-$4m$ volume Thom theorem uses the former, whereas ABS uses a restriction quotient of the latter.