= Solution
For a deterministic $t\le T$, conditional symmetry makes the <conditional characteristic function> of $X_T-X_t$ invariant under $\theta\mapsto-\theta$. The bounded real and imaginary parts of the exponential are legitimate test functions. Hence
$$
e^{-2i\theta X_t}M_t=e^{-i\theta X_t}\mathbb E[e^{i\theta(X_T-X_t)}\mid\mathcal F_t]=e^{-i\theta X_t}\mathbb E[e^{-i\theta(X_T-X_t)}\mid\mathcal F_t]=\mathbb E[e^{-i\theta X_T}\mid\mathcal F_t].
$$
The right side is a bounded complex martingale. Both sides have continuous versions by the assumptions; equality on rational times and continuity make them indistinguishable. Thus \b[$e^{-2i\theta X_t}M_t$ is a martingale] on $[0,T]$. Complex martingale assertions mean the corresponding assertions for both real and imaginary parts.
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