Hi Alex -- In PGA, every operation comes in pairs. There are two exterior products, two inner products, and two geometric products (and the list goes on). If points are represented by vectors, then the quaternion-like sandwich qpq* with the geometric product, where q is now a more general operator in the algebra, always fixes the origin. Thus, it cannot perform Euclidean isometries in regular space because those (in general) move the origin. However, a fixed origin in regular space means that the horizon is fixed in antispace, so if you were to reinterpret vectors as planes instead, then you do get the set of Euclidean isometries that you want. If you had no knowledge of the geometric antiproduct, then you would just say "vectors are planes" and call it a day. That's where plane-based GA comes from. Just use the geometric product and interpret all geometries in antispace instead of regular space. But this throws out the geometric intuition shown in Figures 2.4, 2.5, and 2.7, where vectors, bivectors, and trivectors are simply projected into the w=1 subspace to de-homogenize points, lines, and planes. Furthermore, we need the general notion of product-antiproduct pairs to get things like norms working, anyway, so we might as well use them to avoid dualizing all the geometries.

The space/antispace duality is discussed in Section 2.6, and the fact that the geometric product fixes the origin is discussed in Section 3.5.1. (In case anyone else is wondering, I know ajkjk has a copy of my book.)