Sign symmetry of an ARCH process

With symmetric driving noise whose sign is independent of its magnitude, changing the sign of one driving variable changes the corresponding $X_t$ but leaves future conditional scales unchanged. Consequently $\mathrm{Cov}(X_t,f(X_{t+h}))=0$ for $h>0$ and square-integrable $f(X_{t+h})$. This conclusion uses symmetry, beyond the martingale difference sequence property.