Here's my exact process:

I have the polygon model ready. I go to Create UVs: Automatic Mapping Options. 6 planes, fewer pieces.

Layout is into square, uniform scale mode, bounding box shell stacking, 512 map, .2 percentage space. I hit "Apply", then close the box. I then open the UV Texture Editor. Select the object (not the faces), and rotate it. You can see my problem. The problem shows up even when using a cylinder.