I believe this is the case, but I couldn't come up with a proof off the top of my head, so I want to make sure.

If $A$ is an abelian variety over some field $K$ (I'm in fact interested only in Jacobians, but I don't think this should matter), if $A$ has *some* $m$-torsion point, is the set of all $\sigma(p)$ for $\sigma \in Gal(K)$ and $p$ a $K$-rational $m$-torsion point of $A$, all of the $m$-torsion of $A \times_K \overline{K}$?