techniques
The "inside-out" method for knitting a Moebius loop
Introduction
The inside-out method is currently the popularly preferred method of constructing a Moebius loop, because it avoids the seam of the transverse method. The loop begins with a provisional or provisional-style cast-on on a circular needle; the first half of the first round of knitting is worked on that provisional row as usual, but then the second half of the first round is worked in the bottom of the provisional row. Once that second half is complete, a complete circuit of the boundary edge of the Moebius loop has been completed, and the circular needle is coiled around itself (which is why it is generally recommended that a needle of at least a 40 inch/100 cm length be used). The knitting proceeds around as established by that first round; each round adds to the width of the Moebius loop at its boundary edge, so in fact each round is twice the finished circumference of the loop.
The cast-on can be worked a few different ways. You can choose to use waste yarn, which is somewhat easier, or skip the waste yarn and work your cast-on directly on the circular needle.
The waste yarn method is described by Sarah-Marie Belcastro (text only). A waste yarn method is also described by Daniel Isaksen and Alabama Petrofsky, referencing the methodology taught in 1991 by Maria Iano-Fletcher and Miles Reid, but the directions are not as clearly written as Belcastro's. This waste yarn method is illustrated below.
When using waste yarn in this manner, the type of cast-on used with the waste yarn doesn't matter too much. However, the waste-yarn-less (let's call it "wasteless") version of the inside-out Moebius construction method requires a cast-on that allows you to pick up loops and knit in either direction from the cast-on row without leaving a telltale ridge. One way to accomplish this is by using a backwards loop (half-hitch) cast-on, first knitting into the loops on the needle as normal from the left needle to the right, then sliding the knitting around the circular needle so that the bottoms of each backwards loop can be picked up and knit by the right needle, thus coiling the circular needle into a tight loop. Recognizing that the circular needle cable will inevitably coil on itself, you can combine a pre-coiled needle with the invisible provisional cast-on described by Barbara Walker: instead of using waste yarn, you will work the provisional cast-on over both one end of the circular needle and the cable portion of the needle. A method like this is described in Cat Bordhi's A Treasury of Magical Knitting (but I have to admit that I haven't read any of Bordhi's books). This wasteless method is described on the next page.
However it's started, the inside-out method adapts well to stitch patterns that are designed with at least two rows of garter stitch, because the first round involves knitting into the top of the cast-on, then into the bottom, which generates two rows of garter stitch -- looking at one side, there is a round of knit stitches adjacent a round of purl stitches. (If you continue knitting every stitch, it will stop being garter stitch, and instead will become a band of stockinette that meets a band of reverse stockinette at the middle of the Moebius loop.) It's possible to work around this, of course, but it requires more planning.
Pros and cons
The major disadvantage of the inside-out method is the beginning, because it can be difficult to knit with a coiled circular needle. Once the Moebius loop has grown in width, management of the needle and stitches becomes easier. You may also consider it a disadvantage that extra-long circular needles are needed.
However, if you're familiar with knitting small circumferences with two circular needles, you might find it easier to use the waste yarn cast-on method and two shorter (24 inch/60 cm) circular needles (using the waste yarn method makes it easier to keep track of where you are in the round).
The inside-out method has the advantage of producing a virtually seamless Moebius loop. Sharp eyes will still be able to detect where the loop was started, but it will not be easy (especially if the loop is worked in something other than stockinette, or the yarn has a distracting texture or colour).
Method 1: provisional cast-on using waste yarn
You will need a circular needle of the appropriate needle size, measuring at least 40 inches (100 cm). You may be able to get by with a 32 inch (80 cm) needle, but it will be tight; the key is that you must coil the cable portion into a loop, yet still be able to knit. In addition, you'll need one extra needle: this can be a single double-pointed needle or a straight needle; you just need something for picking up loops so that you can knit them. It doesn't have to be the same size as your circular needle, but preferably it should be the same size or smaller.
You will also need waste yarn, ideally something smooth and in a weight similar to your main yarn, and in a contrasting colour.
1. First, decide how big you want your finished Moebius loop to be, and determine your stitch gauge. Figure out how many stitches you need to cast on to match this size.
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 60 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. With your waste yarn and your circular needle, cast on your number of stitches from step 1. Work as many rows of waste knitting in garter stitch as you wish, back and forth: you'll have a flat piece of knitting on your circular needle in waste yarn. You may even work zero rows, and simply stick with the cast-on row. If so, choose something that has a little body to it -- something other than a backwards loop cast-on (save that particular cast-on method for the wasteless method).
3. Break the waste yarn and switch to your working yarn. Knit one row. This is the first half of the "foundation" round that will define the border of your Moebius loop.
The waste yarn (light blue) was broken, then the main yarn (dark blue) was knit across the live waste yarn stitches.
4. Pretend you're about to join your knitting (without twisting!) for working in the round; the last stitch that was knit should be on the right-hand needle, and your working yarn should be coming from the right-hand needle, too. Flip the fabric on the left-hand needle upwards. You'll see that you've made your moebius half-twist (in the picture, at the bottom left).
Getting ready to knit the second half of the foundation round: note the half-twist introduced in the knitting.
5. Locate the purl bumps of your working yarn, nestled against your contrasting waste yarn. Pick up the first few bumps with your extra needle, then knit them. Note that you're not actually using the left-hand needle of the circular needle to knit; the original stitches are still on the cable of the needle, and the left-hand needle will just dangle.
Picking up stitches to work the second half of the foundation round: the stitches on the blue needle are the purl "bumps".
A number of purl "bumps" have been picked up and knit.
6. Continue picking up and knitting the purl bumps. You'll probably find that you don't need to pick up a series of purl bumps before knitting it, and that you can just use the point of your extra needle to pick up a single bump and knit it with the right needle. Once you are finished picking up all of the purl bumps that are flanked with waste yarn, you should have an odd number of live stitches on your circular needle: 2n-1 stitches, where n is the number of stitches you had originally cast on in waste yarn. If you require an even number of stitches, pick up one more stitch from the strand of your main yarn that leads to the next live stitch on the circular needle, and knit it to make one.
After the purl bumps are picked up, the foundation round is complete and all the stitches to be worked are on the circular needle.
7. Place a marker on the right needle to mark the beginning of the round, if you wish, and continue working in the round on all of the live stitches on the circular needle (the extra needle is no longer needed).
This is a progress photo -- the knitting is halfway through a round of the Moebius loop. Note how the needle is coiled on itself.
8. You can pull out the waste yarn at any time after you have completed the first round. It's possible to pick out the stitches without cutting the waste yarn, but it's easier to snip the waste yarn in the last row before you switched to the main yarn.
Sharp and pointy -- don't cut the fabric, just the waste yarn!
Tips
If you choose your waste yarn judiciously, you can use it as an accent on the finished Moebius loop. For example, plan ahead and knit a ruffle, decreasing to the number of stitches you need for the first round of the main yarn. The ruffle will run around the circumference of the loop, right in the middle, so when the loop is worn half of it will appear to be on the inside. There will be a point (where you started working the second half of the foundation round) where the ruffle stops on one face of the fabric, and starts again on the other face. The waste yarn bit doesn't have to be knit in garter stitch.
My own preference is to use the waste yarn method over the wasteless method, because managing the knitting during the first round seems easier when there is a contrasting colour of yarn holding the stitches in place. It was also possible for me to start knitting in the round with two circular needles right away using the waste yarn method, which I find easier to handle than a single, long, tightly coiled needle. Working without waste yarn is a nifty trick, but for me, it was not a timesaver.
Thanks to Laritza for the photographs and our discussions about methodology!
on to the wasteless inside-out method