= Angular response of a gravitational-wave separation
For a short pair of stationary test particles in <transverse-traceless gauge>, spatial coordinates stay fixed while the measured <proper length> changes. In background signature $+---$, let $h_{xx}=-h_{yy}=H_+$ and $h_{xy}=H_\times$. A coordinate separation of length $\Delta$ at angle $\theta$ has
$$
\ell^2=\Delta^2[1-H_+\cos(2\theta)-H_\times\sin(2\theta)],\qquad
\frac{\delta\ell}{\Delta}=-\frac12[H_+\cos(2\theta)+H_\times\sin(2\theta)]+O(H^2).
$$
The <plus polarization> and <cross polarization> differ by a $45$-degree rotation of their principal axes. With background signature $-+++$ and the same numerical spatial perturbation, the sign reverses because the positive spatial metric is then $\delta_{ij}+h_{ij}$ rather than $\delta_{ij}-h_{ij}$. Consistently redefining the signed strain leaves the physical effect unchanged.
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