  When searching for deviations of statistical isotropy in CMB, a popular
strategy is to write the two-point correlation function (2pcf) as the most
general function of four spherical angles (i.e., two unit vectors) in the
celestial sphere. Then, using a basis of bipolar spherical harmonics,
statistical anisotropy will show up if and only if any coefficient of the
expansion with non-trivial bipolar momentum is detected -- although this
detection will not in general elucidate the origin of the anisotropy. In this
work we show that two new sets of four angles and basis functions exist which
completely specifies the 2pcf, while, at the same time, offering a clearer
geometrical interpretation of the mechanisms generating the signal. Since the
coefficients of these expansions are zero if and only if isotropy holds, they
act as a simple and geometrically motivated null test of statistical isotropy,
with the advantage of allowing cosmic variance to be controlled in a systematic
way. We report the results of the application of these null tests to the latest
temperature data released by the Planck collaboration.
