Aims: In the effort to understand the link between the structure of galaxy clusters and their galaxy populations, we focus on MACS J1206.2-0847 at z ~ 0.44 and probe its substructure in the projected phase space through the spectrophotometric properties of a large number of galaxies from the CLASH-VLT survey. Methods: Our analysis is mainly based on an extensive spectroscopic dataset of 445 member galaxies, mostly acquired with VIMOS at VLT as part of our ESO Large Programme, sampling the cluster out to a radius ~2R200 (4 h70-1 Mpc). We classify 412 galaxies as passive, with strong Hδ absorption (red and blue galaxies), and with emission lines from weak to very strong. A number of tests for substructure detection are applied to analyze the galaxy distribution in the velocity space, in 2D space, and in 3D projected phase-space. Results: Studied in its entirety, the cluster appears as a large-scale relaxed system with a few secondary, minor overdensities in 2D distribution. We detect no velocity gradients or evidence of deviations in local mean velocities. The main feature is the WNW-ESE elongation. The analysis of galaxy populations per spectral class highlights a more complex scenario. The passive galaxies and red strong Hδ galaxies trace the cluster center and the WNW-ESE elongated structure. The red strong Hδ galaxies also mark a secondary, dense peak ~2 h70-1 Mpcat ESE. The emission line galaxies cluster in several loose structures, mostly outside R200. Two of these structures are also detected through our 3D analysis. The observational scenario agrees with MACS J1206.2-0847 having WNW-ESE as the direction of the main cluster accretion, traced by passive galaxies and red strong Hδ galaxies. The red strong Hδ galaxies, interpreted as poststarburst galaxies, date a likely important event 1-2 Gyr before the epoch of observation. The emission line galaxies trace a secondary, ongoing infall where groups are accreted along several directions. Based in large part on data acquired at the ESO VLT (prog.ID 186.A-0798).Full Table 2 is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/579/A4