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. Can always be arranged, although the result may not be ``reduced'' any longer.

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. 

Necessary detail for when $\gcd(\alpha,\beta ) \neq 1;$ if $\langle A,B,C \rangle \equiv \langle P,Q,R \rangle$ by $SL_2(\mathbb Z)$ matrix
 $$  
 \left(  \begin{array}{rr}
  S  &  T  \\
   U   &  V  
\end{array} 
  \right)  ,
  $$
then $\langle -A,B,-C \rangle \equiv \langle -P,Q,-R \rangle$ by $SL_2(\mathbb Z)$ matrix
 $$  
 \left(  \begin{array}{rr}
  -S  &  T  \\
   U   &  -V  
\end{array} 
  \right). $$ In particular, taking the inverse and negating throughout, $\langle -P,Q,-R \rangle \equiv \langle -A,B,-C \rangle$ by $SL_2(\mathbb Z)$ matrix
 $$  
 \left(  \begin{array}{rr}
  V  &  T  \\
   U   &  S  
\end{array} 
  \right).
  $$
What I mean by this is, given Gram (or Hessian) matrix $G$ for $\langle A,B,C \rangle $ and the indicated 2 by 2 matrix, call it W, we get the Gram matrix for the new form by $W^T G W.$