Alright, i have gotten the binary forms part of the story. I may or may not ever know the field side of things, that's life.


From experiments with the website http://www.numbertheory.org/php/classnopos.html and comparison with my own programs, it appeared that the division in half amounted to identifying the distinct forms (whenever the principal form does not represent $-1$)
$$ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{with} \; \; \langle -\alpha, \beta, \gamma \rangle,   $$ where these are ``reduced'' in the sense of Gauss and Lagrange when
$$   \alpha \gamma > 0 \; \; \mbox{and} \; \; \beta > |\alpha - \gamma|.  $$

So it is needed to show that these really are distinct classes when $1$ and $-1$ are distinct as forms. However, this is not hard. If the two forms above are equivalent, then the opposite of $\langle \alpha, \beta, -\gamma \rangle$ in the form class group is $ \langle \gamma, \beta, -\alpha \rangle. $ Thus the hypothesis amounts to
$$ 1 =\langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle.  $$ We are allowed to insist that $\gcd(\alpha,\beta ) = 1,$  important.

The algorithm of Shanks, Buell pages 64-65, tells us that, with   $\gcd(\alpha,\beta ) = 1,$
$$ \langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle = \langle \alpha \gamma, \beta, -1 \rangle.  $$
In short, the hypothesis that $ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{and} \; \; \langle -\alpha, \beta, \gamma \rangle   $ are equivalent implies directly that the principal form represents $-1.$ Very satisfying from my point of view.