Karoubi describes a model of K-theory built on triples: pairs of $C\ell(n+1)$ module bundles $E,F$ with isomorphisms $\alpha:E\rightarrow F$ of their underlying $C\ell(n)$ module bundles. The group of certain equivalence classes of these objects defines the K-theory $K^{-n}$ of the base space.
On the other hand, once one has constructed $K^0$, one can define higher K-theory in terms of $K^0$ as$$\tilde K^{-n}(X)=\tilde K^0(\Sigma^nX).$$Presumably, these notions of K-theory agree. Is there an intuitive way to see why? Can suspending the base space be interpreted as adding Clifford generators that act on the vector bundles?
For example, what do vector bundles on the $n$-sphere have to do with $C\ell(n)$ modules (on a point)? Or do we only see the equivalence of the two K-theories after passing to the stable setting?