Thanks for the tips.  Yes, the grid represents the polygons.  The model started as quads, but I used the subdivide command (then untriangulate).  As a result, I ended up with diamonds instead of squares in many places, with some triangles.  Other triangles were the result of going back and extending some of the vertical and horizontal grid lines through the center of the diamonds, if you know what I mean.

Here is my first attempt to texture my first (successfully) mapped model.  I ultimately want to go for a uniform top for a science fiction setting along the lines of Rocky Jones, Space Ranger.  I used a civil air patrol insignia as the stand-in for the Science and Exploration Agency badge.  I also need a lot of practice painting the UV map, but right now I'm just trying to get the process down.

I'd like to have a lower-polygon version that is used for bending/conforming and a higher-polygon version used for dynamic cloth.  Ideally, they'd use similar texture maps that look identical on the model.