Here is the errant towel #1 - a 5-shaft satin weave. This was woven in 10/2 cotton at 40 epi. Notice the checkerboard on one side and stripes on the other.
Satin weaves are derivatives of twills. The threading on this is a straight draw (1-2-3-4-5) and the treadling is the same (1-2-3-4-5). With a twill tie-up you would get diagonal twill lines but the tie-up for satin scrambles the structure so the diagonals get blurred (although you can still see hints of them in the fabric). Because of the way the twill is scrambled, you need at least 5 shafts to weave a true satin. This one was tied (1-3-5-2-4) and, technically, wove sateen (weft floats) on the top surface and satin (warp floats) on the reverse. It's easier to weave sateen on the top surface because you can do it by lifting only one shaft. If you wanted to weave satin on the top layer you'd have to lift four shafts. Whenever you weave one, the other appears on the reverse side.
The checkerboard shows up on the sateen side because it is weft dominant. The reverse only has hints of the weft color, so it produces warpway stripes instead. If this were sett at 50 epi you would see solid colors - the weft wouldn't show through at all.
My previous post could have been called false damask instead of false satin. Damask is a fabric that has satin and sateen on the same face. To do that you need at least two blocks with at least 5 shafts each, or 10 shafts. That piece was woven with 8 shafts and it scrambles a 4-shaft twill that simulates satin but isn't a true satin weave.
There is a lot more to say about these weave structures, but you'll have to get out your books. Sharon Alderman's Mastering Weave Structures is a good source.