 Solution

The characteristic function is ${\varphi }_X(u)=\mathbb{E}(e^{iuX})$. It satisfies ${\varphi }_{-X}(u)={\varphi }_X(-u)=\overline{{\varphi }_X(u)}$. Thus a real-valued characteristic function gives ${\varphi }_X={\varphi }_{-X}$, and the uniqueness theorem for characteristic functions implies equality of the two distributions. Conversely, equality in distribution gives equality with the complex conjugate, making the characteristic function real. Reality for all $u$ is equivalent to symmetry of the distribution about zero.