Let $V$ be an even-dimensional euclidean vector space, $S$ the spinor module and $E$ a module of the Clifford algebra $C(V)$. Then $$W=\mathrm{Hom}_{C(V)}(S,E)$$ is called the twisting space of $E$. Now here is the issue:
In the book Heat Kernels and Dirac Operators the elements of $W$ are defined to super-commute with the Clifford action$^1$, but I think that they should be defined to commute with Clifford action:
The function $T\in L(S\otimes W,E)$ defined by $$T(s\otimes w)=w(s)$$ is even and hence it super-commutes with the Clifford action if and only if it commutes with the Clifford action. And this is the case if and only if all elements of $W$ commute with the Clifford action:
Fix $x\in C(V)$, $s\in S$ and $w\in W$, then $$(T\circ c_{S\otimes W}(x))(s\otimes w)=T(c_S(x)\otimes 1(s\otimes w))=T(c_S(x)(s)\otimes w)=(w\circ c_S(x))(s)$$and$$(c_E(x)\circ T)(s\otimes w)=(c_E(x)\circ w)(s)$$and hence $T\circ c_{S\otimes W}(x)=c_E(x)\circ T$ if and only if $\forall w\in W:w\circ c_S(x)=c_E(x)\circ w$.
$^1$ To be precise, the elements of $\mathrm{End}_{C(V)}(E)$ are explicitly defined to super-commute with the Clifford action and the exact definition $\mathrm{Hom}_{C(V)}(S,E)$ is not included in the book.