= Solution
The <characteristic function> is $\phi_X(u)=\mathbb E(e^{iuX})$. It satisfies $\phi_{-X}(u)=\phi_X(-u)=\overline{\phi_X(u)}$. Thus a real-valued <characteristic function> gives $\phi_X=\phi_{-X}$, 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. \b[Reality for all $u$ is equivalent to symmetry of the distribution about zero.]
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