Statistical isotropy is supposed to extend beyond the CMB into the matter itself: on sufficiently large scales the galaxy density field and its power spectrum should look the same in every direction, with hemispherical or dipolar variations attributable to sampling variance and survey systematics alone (Peebles 1980; Coles and Lucchin 2002). The matter field is the CMB's descendant, so if one is isotropic the other should be too.
Both keep hinting otherwise, and in the same way. Galaxy and quasar surveys mapped into hemispherical or directional power spectra report asymmetries in clustering amplitude and large-scale power that echo the CMB's hemispherical power asymmetry (Yoon et al. 2014; Appleby and Shafieloo 2014), where one half of the sky carries measurably more fluctuation power than the other. Within ΛCDM the matter-field asymmetry is doubly constrained: it must be sample variance, because the initial conditions are isotropic by assumption, and it must be unrelated to the CMB asymmetry, because the model provides no mechanism connecting a temperature modulation at the last-scattering surface to a clustering modulation in the late-time galaxy field. Two flukes, independently drawn, landing in adjacent directions.
The standing is suggestive rather than decisive: the LSS hints sit at modest significance and survey footprints complicate full-sky statistics. DESI, Euclid, and LSST will deliver the volume and sky coverage to measure directional power at the precision where a real shared axis separates cleanly from two coincident accidents.