Representation theory of SO(1, d-1) and O(1, d-1)

Question

Under a Lorentz transformation, a spinor living in $d$ dimensions transforms as

$\psi (x) \rightarrow \psi'(x') = e^{\frac{1}{2} \lambda^{\mu \nu} \Sigma_{\mu \nu}} \psi (x)$

up to some numerical factors in the exponential from convention. This is only true if we're dealing with the orthochronous proper Lorentz transforms $SO^+ (1,3)$, because the projective spinor representations mean we can just deal with the algebra, and the $\Sigma_{\mu \nu}$ are representations of $so (1, d-1)$. How does this change when we want to think about $P$ and $T$? That is, how does one derive the action of elements of $O (1, d-1)$ on spinors?

The only progress I've made towards understanding this is the coordinate-dependent interpretation put forth in standard texts on QFT, where $P$  and $T$ act with some product of gamma matrices. I was looking for a slightly more general definition.

A related question: is the number of of disconnected components of $O (1, d-1)$ the same for even and odd dimension?

Comments

For $d \geq 2$, $O(1,d-1)$ has always four connected components.