The following discussion is based on the content of FGA explained about the Picard scheme. This is mostly formal: I am trying to find a good way to think about the Poincaré sheaf.
Let us consider $\pi : X\rightarrow S$ a separated $S$-scheme of finite type with a section $\epsilon$. Assume that we are in a situation where the Picard scheme $\mathcal{Pic}_{X/S}$ exists along with a Poincaré sheaf $\mathcal P$, so that it represents the relative Picard functor. This is true for instance when $X$ is a projective abelian scheme over $S$. We assume the Poincaré sheaf $\mathcal P$ on $X\times_S \mathcal{Pic}_{X/S}$ to have a trivialization along $\epsilon\times id$.
Now, applying the universal property of $\mathcal P$ to the invertible sheaf $\mathcal P^{-1}$, I deduce the existence of a unique $S$-morphism $h:\mathcal{Pic}_{X/S}\rightarrow \mathcal{Pic}_{X/S}$ such that for some invertible sheaf $\mathcal N$ on $\mathcal{Pic}_{X/S}$, we have$$\mathcal P^{-1} \cong (id\times h)^{\star}\mathcal P\otimes \pi^{\star}\mathcal N$$
Because $\mathcal P$ and $\mathcal P^{-1}$ both have trivialization along $\epsilon\times id$, the sheaf $\mathcal N$ is trivial, so that $\mathcal P^{-1} \cong (id\times h)^{\star}\mathcal P$.
Is this morphism $h$ simply the inversion in the $S$-group scheme $\mathcal{Pic}_{X/S}$ ?
If it is true, then I can interpret $X\times_S \mathcal{Pic}_{X/S}$ as the base change of $\mathcal{Pic}_{X/S}$ to $X$, and see $id\times h$ as the inversion inside this group scheme over $X$. It would mean that $\mathcal P^{-1} \cong [-1]^{\star}\mathcal P$, that is $\mathcal P$ is a symmetric invertible sheaf.
Is it a correct way to see things ?