On certain solutions of a quadratic form equation

Your ultimate question was answered by Gauss: $O_f^- \cap \operatorname{GL}_2(\mathbb{Z})$ is nonempty if and only if the class of $f$ is ambiguous (i.e. its square is the trivial class).

Indeed, $f(x,y)$ is improperly equivalent to itself if and only if $f(x,y)$ is properly equivalent to $f(y,x)$. As the classes of $f(x,y)$ and $f(y,x)$ are inverses to each other, the claim follows. For more details see Theorem 2.1 in Chapter 14 of Cassels: Rational quadratic forms.

Sections 4 and 6 of the mentioned chapter contain further information on ambiguous classes, e.g. each such class contains a form of type $[A,0,C]$ or $[A,A,C]$, and vice versa.