  In this article we propose a novel test for statistical anisotropy of the
CMB. The test is based on the fact, that the Galactic foregrounds have a
remarcably strong symmetry with respect to their antipodal points and with
respect to the Galactic plane, while the cosmological signal should not be
symmetric or asymmetric under these transitions. We have applied the test for
the octupole component of the WMAP ILC 7 map, by looking at a_3,1 and a_3,3,
and their ratio to a_3,2 both for real and imaginary values. We find abnormal
symmetry of the octupole component at the level of 0.58%, compared to Monte
Carlo simulations. By using the analysis of the phases of the octupole we found
remarkably strong cross-correlations between the phases of kinematic dipole and
ILC 7 octupole, in full agreement with previous results. We further test the
multipole range 2<l<100, by investigating the ratio between the l+m=even and
l+m=odd parts of power spectra. We compare the results to simulations of a
Gaussian random sky, and find significant departure from the statistically
isotropic and homogeneous case, for a very broad range of multipoles. We found
that for the most prominent peaks of our estimator, the phases of the
corresponding harmonics are coherent with phases of the octupole. We believe,
our test would be very useful for detections of various types of residuals of
the foreground and systematic effects at very a broad range of multipoles 2 < l
< 1500-3000 for the forthcoming PLANCK CMB map, before any conclusions about
primordial non-Gaussianity and statistical anisotropy of the CMB.
