I am reading Fulton's but I cannot find a useful result for my problem that seems to be something well known.

Let $\mathcal{J}$ be the Jacobian of a hyperelliptic curve of genus $2$. Consider $D_1,D_2\in \text{Div}(\mathcal{J})$ (two curves inside $\mathcal{J}$) and let $D_2'$ be a translation by a $2$-Torsion point of $\mathcal{J}$.

Q:Is it true that $D_1\cdot D_2=D_1\cdot D_2'$ ?

I cannot see directly that $D_2\sim D_2'$ always. I will appreciate a source in case of being true, or a condition for $D_2$ for this to happen.