techniques
The "outside-in" method for knitting a Moebius loop
Introduction
If you slice up a Moebius loop, you'll get some pretty fantastic results. Cut strips of paper, say 1 inch (2.5 cm) wide by 11 inches (27.5 cm) long, and tape each of them into a loop with a half twist to make Moebius loops and play along.
If you cut a Moebius loop into halves along its circumference, you won't get two pieces, you'll get one extra-long loop with two full twists. The length of the cut you will make will be the circumference of the loop; if you made the loop from an 11-inch-wide strip, your cut length will be 11 inches. (Now cut that extra-long loop in half again and see what you get.)
If you start cutting a loop closer to the edge (say, 1/4 inch or 6 mm from the edge), you'll have to cut twice as long before you reach the beginning of your cut. If you made a loop from an 11-inch-wide strip, you'd have to cut 22 inches to complete the circuit. And then, you won't get a single, extra-long loop; you'll get two linked loops.
The "outside-in" method was postulated by Catherine Kehr and discussed briefly here (scroll to the bottom) by Sarah-Marie Belcastro. It's based on the fact that a Moebius loop with its one face and single edge, cut exactly in half, forms a two-sided loop with two full twists; consequently, a two-sided, two-edged loop with two full twists can be assembled and joined along one of its edges to form a Moebius loop.
Pros and cons
The benefit of this method is that the actual knitting is quite easy: you're knitting in the round along a circumference that measures twice the finished dimension of the Moebius loop, so it's easy to knit something as small as a wristband. There's no troublesome coiling of the circular needle. Also, because you're knitting from the outside inward, there's no need to pick up stitches from the bottom of other stitches; your cast-on row will form the single edge of the Moebius loop.
Another advantage is that this method is easily adaptable to double knitting.
The disadvantage of this method is the finishing. Once you get the hang of it, it's not hard to twist your double-twist loop into a Moebius band; but then, you have to graft, weave, crochet, or otherwise join it together. If you enjoy grafting, then this isn't a huge disadvantage.
Method
1. First, decide on how big you want your finished Moebius loop to be, and determine your stitch gauge. Figure out how many stitches you need to create a loop twice as big.
For example, if you want to make a loop that's roughly head-sized, say 20 inches in circumference, and your yarn knits at 3 stitches per inch, you'll need 120 stitches. (When the Moebius loop is finished, it may measure slightly less than 20 inches due to the thickness of the yarn and the bulk of the fabric at the half-twist. If that's important, you should take that into account when planning your loop.)
2. Cast on that number of stitches you worked out to make the double-long loop on a circular needle that allows you to comfortably work in the round. Join without twisting and work one round. Place a marker to indicate the beginning of the round, if you wish. (There's no point doing the twisting now; you may lose it when you get to the end of the first round.)
3. When you get to the end of that first round, it's time to twist. Make two full (360 degree) twists in the knitting on the needle.
When you're done, the stitches at the needle points should still be "upright" -- you're knitting into these stitches normally, not into their bottoms. In the picture below, the left image shows the knitting is after the completion of the first round, before twisting; the right shows the knitting after two full twists were introduced on the left-hand side.
Add two full twists to your knitting after the first round.
4. Keep knitting complete rounds until you have reached half the final width of your Moebius loop, then slip all stitches to waste yarn.
For example, if you only want the loop to be 4 inches wide, then stop after two inches. Note that if you're going to finish this loop with grafting, you'll be adding one more round of knitting. If that makes a substantial difference to your measurements, then take this into account when deciding when to stop; but keep in mind that if you stop knitting one round earlier, that actually subtracts two rounds from the final width.
The double-twist loop on waste yarn.
5. Mark one of the live stitches (it doesn't have to be at the beginning of the round). Then mark the stitch that is exactly halfway around the loop.
For example, if you have 60 live stitches, mark one stitch. Count 29 more stitches, then mark the 30th stitch. You should have one marked stitch, then 29 unmarked stitches, another marked stitch, and 29 more unmarked stitches.
6. At the first marked stitch, hold the knitting so that the live stitches are uppermost and use a safety pin to mark one side of the fabric. Then, keeping the live stitches at the top, flip the knitting around so that you've still got that same marked stitch, but the other side of the fabric is facing you. Shift the loop around in your hands, keeping this other side of the fabric facing you, until you've reached the other marked stitch. Use another safety pin to mark the side of the fabric facing you, and make sure the safety pin is pointing the same way as the other one.
In other words, if you put the first pin in with the head pointing to your left, put in the second pin with the head pointing to your left. In the pictures below, straight pins were used -- don't use them; they'll fall out.
Stitches and sides marked.
Note that in these pictures, a knit side of the fabric is showing. That's because this loop was worked with exactly half the loop in stockinette and the other half in reverse stockinette. If you had worked this in all stockinette, one side would be purled.
7. Now assemble the Moebius loop. This can be hard, but here are some hints:
Hang the loop around one of your wrists, and arrange the loop so that the live stitches are pointing towards your fingers, and one of the safety pins is facing up on your wrist. The other safety pin should be at the lowest point of the loop as it hangs from your wrist, and it should also be facing up. The live stitches at this other safety pin should be pointing the same way as the stitches at your wrist. If you rest the fingers of your other hand on the lower safety pin, there should be one twist in the loop on either side of your fingers.
Note the direction in which the sides of the loop are twisting. Now, grasping the loop at the lower safety pin, twist the loop in the opposite direction and bring the lower part of the loop over your wrist (pretend that you're doubling a rubber band around your wrist). As you bring the lower part of the loop over your hand, also flip the fabric in your hand over so that the live stitches in your hand butt up against the live stitches resting on your wrist, and the two marked stitches are lined up.
In the end, you want your pin markers to be oriented like this:
Orientation of markers and pins when assembled.
If you attached your safety pin markers so they were pointing in the same direction on opposite faces of the loop, when the Moebius loop is assembled they should be on the same side and pointing in opposite directions.
8. If you did it right, you've got a Moebius loop; it just needs to be patched up. The best way to verify that the loop is assembled correctly is to do a quick basting job. Butt the two live edges together, and sew them loosely together with waste yarn. It will get a little tricky at the point where the Moebius loop twists, but it should work out. If it doesn't work out, you did it wrong; you may have introduced extra twists during assembly, so take out your basting and try again.
The loop, basted together.
9. Now graft (Kitchener stitch) or otherwise join the live stitches together, starting with your marked stitches. You may wind up offsetting your two marked stitches by half a stitch; that's okay, it's the nature of grafting. (It also raises the question of whether your grafting would match perfectly if you had an odd number of live stitches, rather than even; I haven't tried it.)
The live stitches, grafted.
And there you have it, a Moebius loop.
The completed Moebius loop.
Tips
It may help to practice steps 7 and 8 with a paper loop and some adhesive tape first. Make a paper Moebius loop that's big enough to fit over your hand, make a single mark on it that can be assembled in only one direction (sign your name on it), then cut the loop exactly in half, cutting through the mark. Try reassembling this loop and tape it together to verify that you reassembled it the right way.
If possible, trim the waste yarn holding the live stitches as close to the fabric as possible, or use a tapestry needle to weave the waste yarn into the fabric to secure it temporarily, then trim it. Having extra dangling yarn ends can complicate the assembly and grafting steps.
For the same reason, if possible also cut your working yarn before step 7, and also weave the excess yarn into the fabric to hold it out of your way as you assemble the loop. You might not want to cut the working yarn if you're going to use it to graft the loop, though.
As noted above, the loop in the photos was knit half in stockinette, and half in reverse stockinette, so that when the live stitches were butted up against each other the fabric was continuously stockinette or reverse stockinette at that location. Elsewhere in the loop, there is an abrupt change from stockinette to reverse stockinette, but it the change runs transverse to the edge, not parallel to the edge.
If you want to hide all evidence of a purl side, consider working double knitting in the round. The two sides of your double-twist loop will appear as stockinette, and when you graft, you will have to graft twice as many live stitches; however, if you work in a single colour, your Moebius loop will appear even more seamless (so to speak), since there will be no transverse change from knit to purl. If you work double knit in more than one colour, then you will have a colour reversal at the grafting line.