It's area, angles, or distances along great circles through a particular point. (Equivalent, Conformal, and Equidistant respectively in cartography jargon) There are a few other special ones like the Gnomonic projection which maps great circles to straight lines. Stereographic which preserves circles, and Mercator which preserves bearings. (Stereographic and Mercator are both also examples of Conformal projections)
Conformal projections are sometimes described as preserving "shape" but that's not really correct. (How can you preserve shape if you are distorting areas and distances?) They do approximately preserve shape at large scales (If you "zoom in" on a small area, shapes will be approximately correct).
A compass would be completely inappropriate on the maps in question. Consider that "North" for instance is the middle of the topmost square. The only projection on which a compass is appropriate for small scale ("Zoomed out") maps like this is Mercator,
In cartography jargon, your "box sphere" is a collection of 6 different aspects of "gnomonic projection". In cartographic use, other polyhedra are used more commonly, particularly the regular Icosahedron. Even then, polyhedral gnomonic projections like this aren't heavily used as they involve large numbers of inconveniently placed interruptions. More interruptions reduce the distortion within the continuous region of the map, but they are themselves a form of distortion as the surface of a sphere doesn't have discontinuities. Any finite set of discontinuities will never eliminate the distortion in the continuous portions, and as the discontinuities approach infinity, the interrupted map actually approaches an "uninterrupted" projection. (For instance, if you take Interrupted Sinusoidal, and the number of gores (divisions) goes to infinity, the map approaches Equidistant Cylindrical)
A more typical solution to this problem in cartography is the polar inset. You have your main map in one of the typical equatorial aspects, and then include additional sub-maps (insets) in polar azimuthal projections. For instance, a once common way to present maps was a hemispherical map. Two complete hemispheres in equatorial aspects of Stereographic, and then two polar aspect Stereographic insets: http://www.georgeglazer.com/archives...danckerts.html (some dispensed with the polar aspect and just had the two large equatorial ones) You can add polar aspect insets (or insets in any projection) to any map though. For instance, if you had a reference map in Winkel Tripel (Doesn't preserve anything but gives a good balance for an overview), and wanted to show population density data on an inset, you might use an equal area projection for it, like Mollweide or Hammer. Whereas if you wanted an inset for a large scale zoom of a section of the map, you'd want to use a regional projection appropriate to the extent (Transverse Mercator, conic, azimuthal, etc.)
Ultimately you have to decide what you need to preserve, and what you are willing to give up to get it based on the specific requirements of your map. Anyone who claims to have a perfect or universal projection (Like the infamous Gall-Peters) lacks even a rudimentary understanding of cartography, or is lying.