  A pseudo-scalar inflaton field can have interesting phenomenological
signatures associated with parity violation. The existing analyses of these
signatures typically assume statistical isotropy. In the present work we
instead investigate the possibility that a pseudo-scalar inflaton is coupled to
a vector field carrying a small but non-negligible vacuum expectation value
(vev) coherent over our Hubble patch. We show that, in such case, correlators
involving the primordial curvature perturbations and gravitational waves
violate both statistical isotropy and parity symmetry. We compute the Cosmic
Microwave Background (CMB) temperature anisotropies (T) and polarization (E/B)
generated by these primordial modes. The CMB two-point correlation functions
present 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 EB
correlators between $\ell_1$ and $\ell_2 = \ell_1 \pm 2$ multipoles. Such
signatures are specific of the models under consideration and they cannot be
generated if one of parity and isotropy is preserved. As a specific example we
consider 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 a
strong scale-dependent quadrupolar modulation of the primordial power spectra
is generated and we find that almost noiseless data of the large-scale
temperature and E-mode polarization anisotropies (like, e.g., the ones provided
by WMAP or $Planck$) should be able to constrain the quadrupolar amplitude
coefficients $g_{2M}$ of the primordial scalar power spectrum (normalized at
the pivot scale comparable to the present horizon size $k_0^{-1} = 14~{\rm
Gpc}$) down to $g_{2M} = 30$ (68%CL).
