I should say at the outset that I did not develop this technique; I am writing it up here because the explanations online are written either by mathematicians or by knitters, and none of them are particularly clear or simple. I’m neither a mathematician nor a knitter and I’ll leave you to decide if this is clear and simple; it is much easier to demonstrate than to explain.
Moebius scarves are wondrous things. They only have one edge. You can cast them off, but you cannot cast them on…
You need a circular needle. You also need some spare yarn and the yarn you are going to knit with. I strongly recommend getting some chunky yarn and a great big pair of needles for the first time at least, because the work will come on much quicker.
First you use the spare yarn to cast a loose row on to the circular needle. The scarves always come out longer than I expect, so spread out the stitches. When the row stretches across the length of one needle and along the plastic bit but not on to the second needle you change to the yarn you are going to use for the scarf.
Use your main yarn to knit one row into the first row perfectly normally. (In the photo I am treating two yarns as one, the big blue fluffy one and the funky purple one.)
When you get to the end you should be able to hold the circular needle out in a straight line, needles far apart, and have two rows of kintting, the first is the row you cast-on in your spare yarn and the second row is the one you’ve just knitted into it in your real yarn.
This next step is is where you do something very counter-intuitive. If you think about it, a row of yarn is a series of up-loops and down loops, rather like this _n_n_n_n_n_n_ . You normally knit into the top of each loop (the top of the n, if you like) and let the loop you have knitted into fall off the end of the needle. This time you half-twist the row very carefully, and knit into the back of the bottom loops and leave them on the needle.
Wierd, I know, but trust me. By the way, it helps if you are a loose knitter. It is in fact easier to pick these bottom loops up from the circular wire than it is to pick them up from the needle.
The needle you are knitting with (your right needle if you knit right-handed) ends up closer and closer to the piece, with less and less free play but don’t worry about this, it’s just for this one row which closes the loop: you’ll be knitting much more normally as soon as you’ve got round to the end.
When you do get round to the end, you’ve got a very narrow piece of knitting, and a circular needle which is coiled round twice. The cast-on is tucked away in the middle, and there is no bottom edge, just rows of loops on the circular needle. It’s hard to imagine without doing it, but all becomes clear when you try it.
I knot a piece of spare yarn around my right hand needle at this point and switch to knitting purl. When I get round to the piece of knotted yarn I start knitting plain, and so on. People do moss-stitch and garter stitch and ribbing and even lace-work, but I keep it very simple. If you just knit plain, the work will curl, and is much less attractive.
When you run out of yarn, or get bored, or simply can’t wait any longer, cast off. What I ususally do is wait till I’ve run out of yarn, and then use a crochet hook to loop one stitch over another until I’ve fully cast-off, but you can cast-off in any way you feel like. Sew the last few inches of yarn into the work.
What you will have is a moebius scarf, still with the cast-on in the middle which is in the spare wool, and the beginning piece of the yarn you’ve been knitting with. Sew this piece neatly into the work, and use scissors to snip the cast-on out of the piece. I’ve never managed to knit the cast-on in a way which means you can just pull one end and out it comes, though I have heard that it is possible to do so.
Et voila! An impossible scarf.
Oh, and if you like one-sided scarves, you should see the acme klein bottle, its inside is its outside and it is all upside and no downside.