Finally, after weaving two towels, sleying the reed a second time, and weaving another towel with a treadling error, I am weaving a "Turned Taqueté Towel" that I can blog about! The pattern is complicated, but it's actually starting to make sense.
I don't know much about taqueté (a French term with an accent on the last "e"), but I can tell you that it's normally a weft-faced weave. In this case, the weave structure is turned sideways (weft becomes warp), so it's a warp-faced weave. I'm weaving with one shuttle (thankfully): one pic is a pattern weft and one is the tie-down weft. The warp is Cottolin from the Swedish company Borgs Vävgarner, which is half cotton, half linen, and the weft is 20/2 mercerized cotton.
The pattern is a free download from Handwoven magazine. Here's the link, if you'd like to know more: