It is also possible to crochet a Möbius scarf. I'm going to assume that you know some basics of crocheting, at least by reading nocte's writeup. It can be helpful to make a Möbius bracelet or other small object first, just to get a handle on what's actually happening before diving into a more unwieldy project.

The basic process:

  • Chain stitch until you have a length that you like--hold both ends of the chain together and put it around your neck to test, and adjust as necessary.
  • Leave the hook in your last stitch, as though you were about to make another one. Let us pause for a moment to examine the chain you've just made. It has two ends: the end with the tail, where you started, and the end with the hook, where you last worked the yarn. It also has two sides. One of them looks like a row of V shapes or a braid. For simplicity's sake, I'm going to call this the front. The other side looks like a chain of links and bars. I'll similarly call this the back.
  • Lay the chain down on a flat surface, front side up, making sure there are no twists. The hook should still be in your last stitch.
  • Grasp both ends of the chain. Lift them up and bring them together. If you imagine that the loop they make is a very short tube, the front surface will be on the inside.
  • Begin crocheting into the tail end of the chain. You may wonder exactly where to poke your hook into that first stitch. Since you're looking at the back of this particular stitch, it should look something like two curved bits of yarn with a straighter bump in the middle. You only want to bring your new stitches around the outermost loop on the edge you're working, so put your needle between the bump and the curved bit that's farther from you.
  • This first stitch should join the two ends. Because of the way crochet works, this attachment has incorporated a twist. If you look carefully, it should already be a kind of Möbius strip. Had you turned the yarn to stitch into the front side instead, you would have started a tube.
  • Keep crocheting with your favorite stitch, taking care to pull the yarn through only that outer loop for each link in the chain. As you go, you should keep an eye on which part of your work has the single twist, and nudge the twist away from the part you're about to stitch into. This should help you avoid twisting the chain back, working into the front rather than the back, and turning it back into a tube (a common mistake).
  • When you come back to the place where your ends were joined together, you will notice a sort of ledge where the chain begins to have stitches coming out of it. Do you avoid that piece and crochet onto the flat edge? Not unless you want to make a tube! Just keep going over that ledge.
  • The rest should be fairly smooth sailing--you can watch for the tail to count how many half-traversals of the (single) edge you've made so far. When you're satisfied with the width, just finish off as you would any crocheted piece.

Again, I'd recommend practicing on a foundation chain of 15 stitches or so--you can see the entire surface easily and check whether or not you've strayed into tube-land.

The foundation chain will be the "center" of the width of fabric, and each half-traversal of the edge will add a row on alternating sides of it. So, if you're planning on making a pattern of changing stitches or colors, be sure to take that into account. Also, if you're using stitches that have an obvious back and front, one half (starting from the chain and going out to the edge) will be "facing" entirely in one direction, and the other half will be facing the other direction.

Of course, as with the knitted variety, you could simply make a regular scarf, give it a half twist, and join the ends. But that's not nearly as amusing.