Probing the Orientation Distribution of Nearby Double White Dwarfs through Gravitational Waves with LISA

We present a framework to study the orientation distribution of nearby double white dwarf binaries with LISA, explicitly accounting for the fourfold degeneracy in the polarization angle. Exploiting the belt-like sky distribution of the nearby sample along the Galactic plane, we show that the low-multipole extraction can be recast as an effectively one-dimensional Fourier analysis along Galactic longitude, within a spherical-harmonic expansion based on the Wigner-$D$ formalism. The symmetry with respect to the Galactic plane governs the sensitivity: modes even under reflection across the plane remain accessible, while odd modes are strongly suppressed. For a fiducial sample of about 500 binaries within $\lesssim 4$ kpc, corresponding to roughly 5% of the resolved Galactic population, the method yields an uncertainty of order 0.02 for the even coefficients with $\mathrm{\ell }\le 3$, and is readily applicable to the forthcoming LISA catalog.