I am currently reading the book Heat Kernels and Dirac Operators. I think that all Clifford connection should be even such that the associated Dirac operators are odd. (Dirac operators are defined to be odd in definition $3.36$.) It is not clear to me that this a consequence of the definition of Clifford connections (see below). Did they forget to make this part of definition?
Definition $3.32$. A Clifford module $E$ on an even-dimensional Riemannian manifold $M$ is a $\mathbb{Z}_2$-graded bundle $E$ on $M$ with a graded action of the bundle of algebras $C(M)$ on it, which we write as follows: $$(a, s)\mapsto c(a)s,$$where $a\in\Gamma(M, C(M))$ and $s\in\Gamma(M, E)$.
Definition $3.39$. If $\nabla^E$ is a connection on a Clifford module $E$, we say that $\nabla^E$ is a Clifford connection if for any $X,Y\in\Gamma(M, TM)$, $$[\nabla_X^E, c(Y)] = c(\nabla_XY).$$ In this formula, $\nabla$ is the Levi-Civita covariant derivative.