One way to think the reflection principle, which is similar to what you are proposing in your question, is a relation between the index 3 subgroups of $Cl(k^{+})$, which I'll call $I_{3}(m)$, and the subgroups of $Cl(k^{-})$ order 3 which I'll call $S_{3}(-3m)$. It is not difficult to see that $$|S_{3}(-3m)|=\frac{3^{r_{3}^{-}}-1}{2}$$ and that $$|I_{3}(m)|=\frac{3^{r_{3}^{+}}-1}{2},$$ hence any injective map $$ \Phi_{m}: I_{3}(m) \rightarrow S_{3}(-3m)$$ would yield to $r_{3}^{+}\leq r_{3}^{-}$. It is a result of Hasse that the set $I_{3}(m)$ is in bijection with the isomorphism classes of cubic fields of discriminant $m$ (notice that here I'm assuming that $m$ is fundamental, i.e., $m=disc(k^{+})$) hence what we are looking is for a map $\Phi$ that takes a cubic field $K$ and produces a subgroup of $Cl(k^{-})$ of order $3$. In other words given a cubic field $K$ of discriminant $m$ we need to associate a primitive, binary quadratic form of discriminant $-3m$ with the extra condition that the form has order 3 under Gauss composition. To shorten the exposition I'll assume $(3,m)=1$ however all that I'm saying can be worked out in full generality. One natural way to define $\Phi_{m}$ is as follows: Let $O_{K}^{0}$ be the set of integral elements in $K$ with zero trace. Let $q_{K}(x):=Tr(x^{2})/2$. Then, one can show that $q_{K}(x)$ is a primitive, binary quadratic form of discriminant $-3d$. Moreover, as an element of the class group $q_{K}^{2}$ has order $3$. It is possible to show that the map $\Phi_{m}$ sending $K$ to the group generated by $q_{K}^{2}$ is injective, so the result follows.

All the above results should be appearing at some point soon in ANT, but I can email you a copy of the article if you are curious of the details.

**Added:** In response to Alex comment I should say that the other inequality can be also derived with the same ideas I explained above. Now, you start with $I_{3}(-3m)$ and you notice that $$|I_{3}(-3m)|=\frac{3^{r_{3}^{-}}-1}{2}$$ Moreover, $S_{3}(-3(-3m))=S_{3}(m)$ hence by using the trace you get a map $$ \Phi_{-3m}: I_{3}(-3m) \rightarrow S_{3}(m).$$ The diference here is that this map is not injective, but it can be shown that roughly the map is 3-to-1 hence the other inequality. So summarizing Scholz reflection principle is a relation between index 3 subgroups in one class group and subgroups of order 3 in the other, and one way to make this relation explicit is via the trace form.

One place to see where the difference in the behavior $\Phi_{m}$ and $\Phi_{-3m}$ is, as professor Lemmermeyer already pointed out, Bhargava's first paper on Higher composition laws, more specifically Corollary 15. Another place to look at this is J. W. Hoffman and J. Morales, *Arithmetic of binary cubic
forms*, Enseign. Math. (2) **46**, 2000, 61-94.