So, after some more dabbling, in the absence of being able to override an
object's texture, or have a texture applied to a union, etc. be layered over all
other existing textures, it seems that the fundamental issue is getting a
mapping of the  texture intended to be common to all the objects aligned along
the individual components.

I haven't done that much exploring in the texture world, so I see a lot of
experimentation and calculation ahead.

Let's say we have 4 cylinders.  2 upright, and 3 horizontal: |==|
What I'm envisioning is texturing the left side of the "slice" through all of
the objects with one texture, and having all of the remaining stuff on the right
side of the "slice" showing whatever other textures they are defined with.

|=\=|

I'm _guessing_ that if I define a texture and don't translate it, then it's
"centered" around the origin, and when I translate the object itself, it sort of
"travels through a cloud of texturing" in POV-space. Like a gradient x.  (If I
translate the texture, then I slide the texture cloud through POV-space over the
stationary object)  Save me from myself if I'm woefully wrong about this.

Then I rotate it around -z, and I get my "slice" effect.  Hopefully all the
cylinders with the split texture-mapping will have the gradients line up in
register.

The Janet trick is intriguing, though I haven't had much success with blending a
rastered grid texture at [0] to a solid pigment at [1].  A-POVving I shall go...

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