Let $K$ be a local field containing the group $\mu_n$ of $n$th roots of 1 and the $\theta_K:K^*\to G_K^{ab}$ be the reciprocity map. The we know that the Hilbert symbol $$K^*\times K^*\to \mu_n$$ $$(a,b)\mapsto \frac{\theta_K(a)(\sqrt[n]b)}{\sqrt[n]b}$$
is skew symmetric, i.e, $(a,b)=(b,a)^{-1}$; this follows from the fact that $(-a,a)=1$ for all $a\in K^*$.
Now, if $F$ is a formal group over the ring of integers of, say $\mathbb{Z}_p$, and let $\kappa_n$ be the set on $p^n$th torsion points we respec to to the $multiplication-by-p^n$ map $[p^n]=p^nX+\cdots$, then we have the cannonical symbol $$K^*\times M_K^*\to \kappa_n$$ $$(a,b)_{F,n}\mapsto \theta_K(a)(z)\ominus_F z$$ where $M_K$ is the maxiaml ideal of the ring of integers of $K$, $z$ is such that $[p^n](z)=b$ and $\ominus_F$ is subtraction in the formal group $F$. Then, my question is: is there a kind of skew-symmetric property for this symbol?
Is there any type of relation that allows me to interchange the arguments?
