I believe this is what the OP is getting at: the most common weave type for handlebar tape is a "twill weave" where a chevron pattern is prominent. OP's photo is of "plain weave" tape, and you can find Tressostar in plain weave pictured on many sources' sites -- whether it's available that way might be another story, but the pictures don't lie about it being different than a twill. I have Velox, Cateye and Newbaum's around, and every roll is twill. Note to OP: you'll have better luck looking for Velox Tressostar if you spell it correctly, though I know that correct spelling is fast falling out of fashion.

I believe that a twill is woven by taking the weft threads (the ones running the length of the ribbon) over two warp threads and under one, and then for the next warp thread you move the weft sequence up one warp thread each time. Then, to make the chevron, when you get to the middle of the tape, you start going back the other way, moving the weft sequence down a warp thread every time, rather than up.



Plain weave, OTOH, has the weft going over one warp thread, and under one, with successive weft threads alternating the over and under warp threads.



By the way, there's no woven tape that is "non-weave" (doh!) -- if you want that, then you either have to use some crazy type of weave, very fine, like sateen (such as ribbon for presents), or else go to a leather or synthetic (polymer or plastic) tapes like Benotto(which seems like sateen to me) or cushier PVA or some other plastic, with a pattern, dimples, or with holes. There are a lot of these, and they have pros and cons.