  We introduce new symmetry-based methods to test for isotropy in cosmic
microwave background radiation. Each angular multipole is factored into unique
products of power eigenvectors, related multipoles and singular values that
provide 2 new rotationally invariant measures mode by mode. The power entropy
and directional entropy are new tests of randomness that are independent of the
usual CMB power. Simulated galactic plane contamination is readily identified,
and the new procedures mesh perfectly with linear transformations employed for
windowed-sky analysis. The ILC -WMAP data maps show 7 axes well aligned with
one another and the direction Virgo. Parameter free statistics find 12
independent cases of extraordinary axial alignment, low power entropy, or both
having 5% probability or lower in an isotropic distribution. Isotropy of the
ILC maps is ruled out to confidence levels of better than 99.9%, whether or not
coincidences with other puzzles coming from the Virgo axis are included. Our
work shows that anisotropy is not confined to the low l region, but extends
over a much larger l range.
