  We begin showing that for even dimensional vector spaces $V$ all
automorphisms of their Clifford algebras are inner. So all orthogonal
transformations of $V$ are restrictions to $V$ of inner automorphisms of the
algebra. Thus under orthogonal transformations $P$ and $T$ - space and time
reversal - all algebra elements, including vectors $v$ and spinors $\varphi$,
transform as $v \to x v x^{-1}$ and $\varphi \to x \varphi x^{-1}$ for some
algebra element $x$. We show that while under combined $PT$ spinor $\varphi \to
x \varphi x^{-1}$ remain in its spinor space, under $P$ or $T$ separately
$\varphi$ goes to a 'different' spinor space and may have opposite chirality.
We conclude with a preliminary characterization of inner automorphisms with
respect to their property to change, or not, spinor spaces.
