A little late, but maybe this is useful
To make top and bottom up versions (= otherwise the same projection, but 180 degrees differece in X-rotations) identical in Illustrator's 3D extrusion the shapes cannot be the same. The other shape must be = the first shape as flipped around horizontal axis.
Here the X-rotations are 75 degrees and -105 degrees. The results are identical.
About "does Illustrator have 3D?" Illustrator can show how a rendered 3D extrusion or revolution surface would look out. But there's no accessible 3D surface. You cannot build proper combined pieces. An example:
In the left there's 2 extruded shapes. This is in a 3D CAD program. Illustrator would make the same pieces in few seconds.
In the middle the extrusion results are moved to overlap in 3D space. In Illustrator you cannot do this, because there's no accessible 3D pieces, you have only the 2D images of the rendered extrusion results. To make the same scene in Illustrator you must cut a hole to one of the shapes.
In the right there's a boolean difference of 3D shapes. It was 2 clicks in 3D CAD. Illustrator has nothing to make it in 3D, you must draw it.
About vector rendering of a 3D scene: It's sometimes useful to export the 3D scene as 2D vector wireframe and open it in Illustrator:
In this way it's well possible to get some tricky 2D drawing work done with very little effort. In Illustrator one can start from coloring or drawing the easy pieces.
About artwork mapping: It's a calculated 2D image how a 3D piece with mapped images would look out.
The mapping often is impossible because Illustrator divides surfaces to pieces. Only very simple shapes can be extruded or revolved so that you get the expected number of surfaces. Generally you may think you need 1...4 art symbols because you have 1...4 visible textured surfaces. But if the surfaces are divided, you must cover separately for example a dozen of them. Making all to fit is a nightmare.