  We propose and develop a formalism to describe and constrain statistically
anisotropic primordial perturbations. Starting from a decomposition of the
primordial power spectrum in spherical harmonics, we find how the temperature
fluctuations observed in the CMB sky are directly related to the coefficients
in this harmonic expansion. Although the angular power spectrum does not
discriminate between statistically isotropic and anisotropic perturbations, it
is possible to define analogous quadratic estimators that are direct measures
of statistical anisotropy. As a simple illustration of our formalism we test
for the existence of a preferred direction in the primordial perturbations
using full-sky CMB maps. We do not find significant evidence supporting the
existence of a dipole component in the primordial spectrum.
