This is a nice question, and I look forward to seeing a definitive answer.  Meanwhile, let me point out that it is known that a convex polytope in $E^n$ is not always determined by a *finite* set of its projections, but it is determined by *all* of its 2D projections.  Richard Gardner, in *Geometric Tomography*,
puts the first point this way: "it is generally not possible to choose a finite set of subspaces in such a way that the corresponding projections distinguish $P$ from every other convex polytope" [p.93].