= Solution
Write $F_x(\mathbf h)=f_x S_x(h,k,l)$, where the four-site orbit has <structure factor>
$$
\begin{aligned}
S_x(h,k,l)={}&e^{2\pi ix(h+k+l)}+(-1)^{h+l}e^{2\pi ix(-h-k+l)}\\
&+(-1)^{k+l}e^{2\pi ix(-h+k-l)}+(-1)^{h+k}e^{2\pi ix(h-k-l)}.
\end{aligned}
$$
A cyclic permutation of $h,k,l$ fixes the first term and permutes the remaining three. Thus $S_x(k,l,h)=S_x(h,k,l)$. Substituting $-h,-k,l$ exchanges the first and second exponentials and the third and fourth, with the common coefficient $(-1)^{h+l}$:
$$
\boxed{S_x(-h,-k,l)=e^{-i\pi(h+l)}S_x(h,k,l)}.
$$
The two forms of this phase, $e^{+i\pi(h+l)}$ and $e^{-i\pi(h+l)}$, agree for integer indices. Multiplication by a unit-modulus phase preserves intensity, irrespective of whether $f_x$ is real or complex. Since the phase is independent of $x$ and atomic species, it also multiplies the total <structure factor> of the crystal.
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