Characteristic function of symmetric distribution

Question

I know that if $X$ is a symmetric random variable, then the characteristic function is real, since $\overline{E[e^{itX}]} = E[e^{-itX}]=E[e^{itX}]$. However, is there a simple way (without appealing to some sort of inversion) to show the converse? (If the characteristic function of a distribution is real, then the distribution is symmetric.)