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.