Contrary to frequent statements, the indications from WMAP and Planck data analyses suggesting the possible existence of a privileged space direction are not strange phenomena from the point of view of fundamental space-time geometry. As already emphasized in our previous papers on the subject, the spinorial space-time we suggested in 1996 naturally incorporates a privileged space direction for each comoving observer. This space direction is simply defined by the sigma-like matrix of which the cosmic space-time position spinor is an eigenspinor. In this series of papers, we remind some of our previous work and discuss the present situation.