What is the transformation law for spinors (pinors) under parity in an odd number of spacetime dimensions?
I know how to derive the transformation properties of spinors (pinors) under parity in an even number of spacetime dimensions. Let $$\eta^{ab} = \mathrm{diag} (1, 1\ldots 1, -1, -1 \ldots -1) $$
where there are $p$ entries of $1$ and $q$ entries of $-1$, and $p+q=n$. Let the gamma matrices $\gamma^a$ generate the real Clifford algebra $\mathrm{Cl}(p,q)$, $$ \{ \gamma^a, \gamma^b \} = 2 \eta^{ab} $$
The Pin group $\mathrm{Pin}(p,q)$ is defined as the set of invertible elements $S_{\Lambda}$ of $\mathrm{Cl}(p,q)$ that satisfy $$ S_{\Lambda} \gamma^a S_{\Lambda}^{-1} = {\Lambda^a}_b \gamma^b $$
for some element ${\Lambda^a}_b$ of the orthogonal group $\mathrm{O}(p,q)$, and also $S_{\Lambda}S_{\Lambda}^{\tau} = \pm 1$, where the superscript $\tau$ denotes a linear operator that reverses the order of products, e.g. $(\gamma_0 \gamma_1 \gamma_2)^{\tau} = \gamma_2 \gamma_1 \gamma_0$. For each $\Lambda$, there are two solutions for $S_{\Lambda}$ that differ by a minus sign, and the the map that sends these two solutions to $\Lambda$ is a $2-1$, homomorphism from $\mathrm{Pin}(p,q)$ to $\mathrm{O}(p,q)$.
A parity transformation in the orthogonal group $\mathrm{O}(p,q)$ that inverts the $i$-th spatial axis is given by $P_i = \mathrm{diag}(1, 1 \ldots 1, -1, 1, 1, \ldots 1)$, with the entry of $-1$ acting on the $i$-th spatial axis. To find a parity transformation on a spinor (pinor), one solves the above equations for $\Lambda = P_i$. In an even number of spacetime dimensions, the solution is $$ S_{P_i} = \pm \gamma_i \gamma_n $$ where $\gamma_n = \prod_{a=0}^{n-1} \gamma_a$.
Crucial to make this work is the fact that $\gamma_n$ anti-commutes with all of the gamma matrices $\gamma_a$ in an even number $n$ of spacetime dimensions. However, in an odd number of spacetime dimensions, the operator $\gamma_n$ is proportional to the identity matrix, and thus commutes with everything. And indeed I believe there is no operator in an odd number of spacetime dimensions that anticommutes with all the gamma matrices $\gamma_a$. As a result, I cannot see that there is a solution to the above equations for a parity transformation on spinors (pinors) in odd dimensions.
The main reference I have been using is http://arxiv.org/abs/math-ph/0012006. In section 5, page 65, a similar conclusion is reached. Then it is said that the 2-1 homomorphism/covering map from $\mathrm{Pin}(p,q)$ to $\mathrm{O}(p,q)$ given above is not surjective in an odd number of spacetime dimensions, and in particular it does not 'hit' axis reflections in $\mathrm{O}(p,q)$.
