Let $X$ be a smooth projective irreducible curve over an algebraically closed field $k$, and let $\omega$ be a rational 1-form on $X$, so $\sum_x {\rm{res}}_x(\omega) = 0$ (sum over $x \in X(k)$). For the collection of residues ${\rm{res}}_x(\omega)$ indexed by the finite set of poles of $\omega$, we can form the "second symmetric function" $$s_2(\omega) = \sum_{x,x'} {\rm{res}}_x(\omega){\rm{res}}_{x'}(\omega)$$
of these residues, where the sum is indexed by *unordered* pairs of poles. Note that in characteristic 2 the contribution from the terms $x'=x$ is $\sum {\rm{res}}_x(\omega)^2 = (\sum {\rm{res}}_x(\omega))^2 = 0$, so if you prefer to omit the "diagonal" contribution then that's fine (as the motivating situation is in characteristic 2). One can just as well consider higher symmetric functions in the residues.

The question is: has such a construction been studied somewhere in the literature, or more specifically does $s_2(\omega)$ have any interesting properties? The motivating case of interest is when ${\rm{char}}(k)=2$ (in which case $p_1^{\ast}(\omega) \wedge p_2^{\ast}(\omega)$ on $X \times X$ descends to the 2nd symmetric power $X^{[2]}$of $X$, so perhaps in such cases one can find a link with residue symbols for this descended rational 2-form on $X^{[2]}$...) and $\omega$ has only simple poles (and moreover $\omega$ varies in a certain controlled manner).