  As well as the galaxy number density and peculiar velocity, the galaxy
intrinsic alignment can be used to test the cosmic isotropy. We study
distinctive impacts of the isotropy breaking on the configuration-space
two-point correlation functions (2PCFs) composed of the spin-2 galaxy
ellipticity field. For this purpose, we build a formalism for general types of
the isotropy-violating 2PCFs and a methodology to efficiently compute them by
generalizing the polypolar spherical harmonic decomposition approach to the
spin-weighted version. As a demonstration, we analyze the 2PCFs when the matter
power spectrum has a well-known $g_*$-type isotropy-breaking term (induced by,
e.g., dark vector fields). We then confirm that some anisotropic distortions
indeed appear in the 2PCFs and their shapes rely on a preferred direction
causing the isotropy violation, $\hat{d}$. Such a feature can be a distinctive
indicator for testing the cosmic isotropy. Comparing the isotropy-violating
2PCFs computed with and without the plane parallel (PP) approximation, we find
that, depending on $\hat{d}$, the PP approximation is no longer valid when an
opening angle between the directions towards target galaxies is ${\cal
O}(1^\circ)$ for the density-ellipticity and velocity-ellipticity cross
correlations and around $10^\circ$ for the ellipticity auto correlation. This
suggests that an accurate test for the cosmic isotropy requires the formulation
of the 2PCF without relying on the PP approximation.
