As Piotr has suggested, Torelli answers the question if the canonical principal polarization is included in the data.

If $C$ is a genus $g$ curve, then consider the pair $(X,\Theta)$, where $X=Pic^{g-1}_C$ and $\Theta$ is the locus of effective classes, and the pair $(A,\lambda)$, where $A=Pic^0_C$ and $\lambda$ is the associated canonical pp. Giving $(X,\Theta)$ over $K$ is equivalent to giving $(A,\lambda)$ over $K$. [More generally, there is an isomorphism of stacks from the stack of suitable pairs $(X,\Theta)$, where $X$ is a torsor under an abelian variety $A=Aut^0_X$, $\Theta$ is an ample effective divisor on $X$ such that $\dim H^0(X,\Theta)=1$ and there is an involution $\iota$ of $X$ that preserves $\Theta$ to the stack of ppav $(A,\lambda)$.] If $C$ is non-hyperelliptic then Andreotti's proof of Torelli recovers $C$ from $\Theta$: the branch divisor $B$ in $\mathbb P^{g-1}$ of the Gauss map $\Theta -\to \mathbb P^{g-1}$ is the projective dual of $C$. According to the classical biduality theorem, the dual of $B$ is $C$. The biduality theorem does not always hold in char. $p>0$, but Andreotti shows that for canonically embedded curves it does hold (with some strange behaviour in char. $3$ for twists of the Fermat quartic). In particular, $C$ is defined over $K$ and $X$ is $K$-isomorphic to $Pic^{g-1}_C$, with the theta divisors matching.

As Piotr has suggested, Torelli essentially answers the question if the canonical principal polarization is included in the data. More accurately, this is a version of Torelli due to Serre: if $(A,\lambda)$ is a ppav of dimension at least $2$ over a field $K$ and if over some extension $L$ of $K$ there is a curve $C$ such that $(A,\lambda)_L$ is $L$-isomorphic to the Jacobian of $C$, then there is a curve $C_0$ over $K$ such that $C$ is isomorphic to $(C_0)_L$ and some quadratic twist of $(A,\lambda)$ is $K$-isomorphic to the Jacobian of $C_0$. Moreover, if $C$ is hyperelliptic then no twist is necessary.