I hope someone can help me, although this question is rather specific.

I am reading John Roe's chapter on Getzler symbols in "Elliptic operators, topology and asymptotic methods" to understand the proof of the Atiyah-Singer index theorem.

- So, for the differential operators on functions of $M$ (say, $D(M)$), there is a symbol map to the space of constant coefficient operators (regarding a cartesion chart) on $T_p M$, named $C(TM)$. (This map seems quite natural, because $C_m(TM)$ is isomorphic to $D_m(M)/D_{m-1}(M)$)
- By the same principle, there is a symbol map from the Clifford algebra $Cl(TM)$ to the exterior algebra $\Lambda^\bullet TM$.

Now regarding differential operators on the spinor bundle $\Sigma M$, we have $D(\Sigma M)=D(M) \otimes Cl(TM)$.

**And so I thought:**

Looking at the above, I would consider the symbol map from $D(\Sigma M)$ to sections of $C(TM) \otimes \Lambda^\bullet TM$. But that is not the one constructed in the Getzer calculous.

Rather, the symbol map used there maps $D(\Sigma M)$ to sections of $P(TM) \otimes \Lambda^\bullet TM$, where $P(TM) = \mathbb{C}[TM]\otimes C(TM)$, the space of constant coefficient operators on $TM$ with polynomial coefficients.

**And here is my question:**

Why is that? I guess, this construction is just the ingenuity of the whole construction. But can someone give me a motivation? What does this symbol map do, what the one I would have thought of cannot do?