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 Binary Quadratic Forms, 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. Note that we do not need to use Shanks, various books discuss the ``united forms'' approach of Dirichlet, pages 55-57 in Buell. On page 57, he confirms, with $\color{green}{\gcd(a_1,a_2,B)= 1},$ that $$\color{magenta}{ \langle a_1, B, a_2C \rangle \circ \langle a_2, B, a_1C \rangle = \langle a_1 a_2,B, C \rangle}. $$ Dirichlet gives the same outcome as the Shanks method, but with no additional $\gcd$ assumptions, using $$\color{magenta}{ a_1 = \alpha, a_2 = \gamma, B = \beta, C = -1.} $$ This is also Theorem 98 on page 138 of Leonard Eugene Dickson, Introduction to the Theory of Numbers.