There weren't any modifiers on the mesh to apply in this case.  Besides, I don't see how not applying a modifier to the mesh (had I used them) would cause the UVMap to become twisted.  Wouldn't not applying modifiers and removing them from the stack (such as Mirror or Solidify) cause the exported mesh to itself be deformed or incomplete?  The mesh itself was not deformed, it imported to Poser exactly as I had created it in Blender.  That is, the shape was correct, the vertices were all where they should be and so forth.  The UVMap, in specific locations, simply wasn't being applied the way it had been mapped.  For example on the front of the base the texture was turned at what looked like about 30 degree CCW, and somewhat skewed, but the sides and back were fine.  The middle portion of the orb was fine, but the very top and bottom edge (cooresponding to the last rows of polygons) were inexplicably stretched and skewed.   Whatever the cause, as you can see from the render above they are all fine now.

I suspect that while I was unwrapping and editing the UVMap I must have hit a hot key such as F or something causing it to create connections in the mesh between vertices that shouldn't have been there and which I couldn't see an that may have produced the warping effect.  When I rebuilt the mesh from an earlier model (which didn't exhibit the problem), I was very careful about making sure I didn't do that accidentally, and things seem fine now. 

There was one other issue that was resolved by subdividing a few square facets into two triangles.  Poser seems to prefer triangular facets to square facets in some cases (if the square is stretched into a trapazoid for example).  Or at least, when I converted the problem facets to triangular ones the remaining distortion disappeared.

I will keep your note about modifiers in mind in the future.  I can imagine that could cause quite a lot of things to go wonky.