Generalized identifiability of sums of squares

Let $f$ be a homogeneous polynomial of even degree $d$. We study the decompositions $f={\sum }_{i=1}^r{f}_i^2$ where $\mathrm{d}\mathrm{e}\mathrm{g}{f}_i=d\mathrm{/}2$. The minimal number of summands $r$ is called the $2$-rank of $f$, so that the polynomials having $2$-rank equal to $1$ are exactly the squares. Such decompositions are never unique and they are divided into $\mathrm{O}(r)$-orbits, the problem becomes counting how many different $\mathrm{O}(r)$-orbits of decomposition exist. We say that $f$ is $\mathrm{O}(r)$-identifiable if there is a unique $\mathrm{O}(r)$-orbit. We give sufficient conditions for generic and specific $\mathrm{O}(r)$-identifiability. Moreover, we show the generic $\mathrm{O}(r)$-identifiability of ternary forms.