Moebius cowl, finished
Cat Bordhi demonstrates her mind-blowing Moebius Cast-On here: ca.youtube.com/watch?v=LVnTda7F2V4
The cast-on becomes the MIDDLE of the cowl, and the needle goes all the way around the whole thing. I don't know the right terminology, but if this was a tube you'd have the needle going around the circumference twice. Of course, since this is a moebius, it's all one edge!
If you knit all the way around, whatever bit of the cowl you look at, the fabric on one side of the cast-on will look like knitting/stockinette, and the other side will look like purling/reverse stockinette. In this case, I knit around the loop 5 times, then purled around 5 times, to a total of 25 rows. That looks like knit ridge, purl ridge, knit, purl, knit on one side of the cast-on, and purl, knit, purl, knit, purl on the other. (For those who can't "read" knitting, the purl ridges are the ones bulging out towards you, and the knit ridges curl away, here, so you're mostly seeing the 5 purl ridges...)
Finally, you cast off -- in this case, with an applied i-cord edge, which creates a small tube of stitches that run perpendicular to the edge. Be warned: with a moebius, the cast-off takes nearly as much time as the rest of the knitting put together!