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A pseudo-scalar inflaton field can have interesting phenomenologicalsignatures associated with parity violation. The existing analyses of thesesignatures typically assume statistical isotropy. In the present work weinstead investigate the possibility that a pseudo-scalar inflaton is coupled toa vector field carrying a small but non-negligible vacuum expectation value(vev) coherent over our Hubble patch. We show that, in such case, correlatorsinvolving the primordial curvature perturbations and gravitational wavesviolate both statistical isotropy and parity symmetry. We compute the CosmicMicrowave Background (CMB) temperature anisotropies (T) and polarization (E/B)generated by these primordial modes. The CMB two-point correlation functionspresent distinct signals of broken rotational and parity invariance.Specifically, we find non-vanishing TT, TE, EE and BB correlators between$\ell_1$ and $\ell_2 = \ell_1 \pm 1$ multipoles, and non-vanishing TB and EBcorrelators between $\ell_1$ and $\ell_2 = \ell_1 \pm 2$ multipoles. Suchsignatures are specific of the models under consideration and they cannot begenerated if one of parity and isotropy is preserved. As a specific example weconsider the simple case in which the vector field has just an "electric"background component decaying in the standard way as $a^{-2}$. In this case astrong scale-dependent quadrupolar modulation of the primordial power spectrais generated and we find that almost noiseless data of the large-scaletemperature and E-mode polarization anisotropies (like, e.g., the ones providedby WMAP or $Planck$) should be able to constrain the quadrupolar amplitudecoefficients $g_{2M}$ of the primordial scalar power spectrum (normalized atthe pivot scale comparable to the present horizon size $k_0^{-1} = 14~{\rmGpc}$) down to $g_{2M} = 30$ (68%CL).

Parity-violating and anisotropic correlations in pseudoscalar inflation