Automorphisms of $SO(n,\mathbb R)$
See Outer automorphism group wiki page, the section on real Lie groups. It says that outer automorphism groups are symmetries of Dynkin diagram.
From this it follows that for $SO(2n-1, \mathbb{R})$, i.e. series $B_n$, all automorphisms are inner. For $SO(2n, \mathbb{R})$ there is order 2 outer automorphism which indeed coincides with conjugation by reflections.
So it follows that the answer to your question is in affirmative, and $C \in SO(n, \mathbb{R})$ for odd $n$, and in $O(n, \mathbb{R})$ for even.