Can a random variable symmetric about zero be uniquely defined by the difference of two other iid random variables with mean zero?

Question

Let $Y$ be a random variable symmetric about $0$ with characteristic function $\phi_Y(t)$.

If $Y=X_j-X_i$, with $X_i,X_j \overset{iid}{\sim} X$, with characteristic function $\phi_X(t)$.

Then $\phi_Y(t)=\phi_X(t)\phi_{-X}(t)=\phi_X(t)\phi_{X}(-t)=\phi_X(t)\overline{\phi_X(t)}$.

With the constraint $E[X]=0$, i.e., $i^{-1}\phi_X^\prime(0)=0$, is $\phi_X(t)$ unique?

For example, if $Y \sim \mathcal{N}(0,2)$, is $X \sim \mathcal{N}(0, 1)$ unique?