There are not enough words to go around, so i invented a lot. The basic notion of space and subspace is that it is a "fabric" (ie -ix), from which you cut "patches" (-on). So a "hedron" is a 2d patch, of which "poly" is (many with closure). A "poly hedron" is then "many 2d patches with closure".
One then has names for 2d fabric in any dimension, eg "hedrix". Plane suggests division (cf plain = what you stand on). The idea is that the terminology is consistant in higher dimensions, and therefore each stem must be applied to one meaning only, and not to several.
Part of the richness of the PG is that there are explicit words for, say the surface of an 5d sphere, regardless of whether this is S4, or any 5d sphere (in 5 or higher dimensions). The surface is a "glomoterix" (globe-shaped 4-fabric), and the body is a "glomoteron" (globe-shaped 4-patch). Only patches bound.
Note also we can describe a tiling of patches, by using 'aperi' (a = without), peri = bounds of referenced points in a space. So a tiling of hexagons (2-patches or 'hedra') is an aperihedron.
Much of the confusion with the 'tiger' (tri-circular torus), is because the thinking that the 2d margin (ie N-4 fabric), somehow divides space, and therefore can not be enclosed.
You can indeed produce a map of directions in 3d, of all 4d vectors, by using the same sorts of mapping that render S2 onto E2. You can, for example, use azithmal projection (projected through the centre of a sphere, giving a half-sphere, but preserving great circles to lines), or stereographic (preserves angles, the centre of projection is the opposite pole), or orthogonal.
Projections like mercartor's do not make much sense, because the nature of rotation is different in four dimensions, to three.
Clifford-parallels does in fact correspond to hamilton's quarterions. If you pick a quarterion, say q, the rotation from 1 to q, will rotate every point by an identical angle, from x to qx. You can reverse the all-rotation, and make it into a great-arrow rotation, by using x => qxq'.
Great arrows are great circles with directions on them.
PG is still on line at http://www.geocities.com/os2fan2/gloss/index.html