I was previously under the misapprehension that time $T$ and parity $P$ symmetries in conjunction ($PT$) were a reflection in $(3+1)$-dimensional space-time, where
$$P: \vec x \to -\vec x$$ $$T: t \to -t$$
but, in fact, because it has determinant $(-1)^4=1$, $PT$ is merely a rotation (by $\pi$). Is this correct? Shouldn't the symmetry we seek be a reflection in space time, to generalize parity so that it includes time?
In other words, if $P$ is a reflection in $3$ spatial dimensions, why isn't $PT$ defined to be a reflection in $(3+1)$-dimensional space-time?
My thought is that $P$ is an intuitive symmetry in spatial dimensions. In the spirit of relativity, why don't we generalize $P$ from a reflection in space to a reflection in space-time (called, say, $PT$)? Why are $P$ and $T$ reflections considered separately? It's problematic to me because their combination ($PT$) is not a reflection. I suppose I know that $P$ and $T$ are separate because only $T$ must be anti-unitary etc...