= Solution
Moving the origin by $\mathbf t$ changes a structure-factor phase by $-2\pi\mathbf h\cdot\mathbf t$ in the corresponding coordinate convention. The triplet phase
$$
\phi_{\mathbf h}+\phi_{\mathbf k}-\phi_{\mathbf h+\mathbf k}
$$
is unchanged because the three origin shifts cancel. In a centrosymmetric origin convention, the associated sign product $S(\mathbf h)S(\mathbf k)S(\mathbf h+\mathbf k)$ is therefore invariant.
For example shifting $x$ by half a period flips both $S(10)$ and $S(11)$, leaving $S(01)$ unchanged; shifting $y$ flips the other corresponding pair. A general origin may make the factors complex, but the phase-invariant statement remains valid.
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