techniques
The "inside-out" method for knitting a Moebius loop
Method 2: provisional cast-on without using waste yarn
For this "wasteless" method, you'll 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.
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. Coil your circular needle as shown in the picture and hold the coiled needle in your right hand: your hand should be holding both the right-hand needle and part of the cable loop. You will be casting on stitches over the right-hand needle tip and the needle cable. (These instructions are written for a "right-handed" knitter who knits stitches off the left-hand needle and onto the right-hand needle.)
3. Make a slip knot with your working yarn and place it on the right needle. Hold both the right-hand needle and the needle cable in your right hand as shown; note that the cable is in front, and the yarn is in back.
The slipknot is placed on the needle to be used for casting on; the yarn is held behind the needle cable.
4. Start casting on using the invisible provisional cast-on method and the right-hand needle, treating the needle cable as the waste yarn.
The working yarn is more or less wound in a "figure 8" path around the needle and the cable.
Continue casting on until you have your target number of stitches cast onto the right-hand needle. Because this is a provisional cast-on, for every pass over the right-hand needle, you'll have a corresponding pass under the cable. This will give you a total of two times your target number of stitches.
For example, if you figured out that your wearing circumference required 60 stitches, you'd cast on until you had 60 loops over the right-hand needle and 60 loops around the cable. This equals 120 live loops.
When you're finished, depending on the size of your cable needle and the number of stitches cast on, the stitches may be riding up towards the other needle, as you can see in the photo below. Your slip knot will be sitting on the cable, not the left-hand needle.
The cast-on stitches cover the right-hand needle shaft and the cable on the right; the doubled cable (including a crossover, hidden here) in the middle; and the cable and remaining needle shaft on the left.
5. Once you have finished casting on, keep the needle you were using for casting on in your right hand, and pick up the other needle with your left hand. You're ready to start knitting the first round of the Moebius loop. Move the stitches around the coiled cable needle until you have loops ready to knit at the point of the left-hand needle. (Remember, when you had cast on using the right-hand needle, half of your stitches were on the needle, and the other half were on the cable. Because you coiled your cable around, those stitches that were cast onto the cable will be the first ones to be knit from the left-hand needle.)
Place a marker to indicate the beginning of the round, if you wish, and start knitting. Because of the way the stitches were cast onto the needle, you will find that you have to alternate knitting in the back and knitting in the front of the provisionally cast-on stitches in order to create untwisted stockinette. Each round will equal twice the target number of stitches you calculated. After you have worked the first round of the Moebius loop, it will no longer be necessary to alternately knit into the front and back to "correct" the stitch mounts.
Tips
When you're finished casting on, you might find that the stitches hold the needles rather snugly, making it difficult to get the needle points into the correct position for knitting. You may have to gently enlarge the coiled needle cable, or redistribute the slack in the cable, in order to move the points into a comfortable position. If you're using interchangeable needles, do it carefully--this manipulation can put a lot of stress on the joins.
This method results in an even number of stitches around the boundary edge of the Moebius loop. If you need an odd number of stitches, it's easy to increase after the provisional cast-on: just bring the yarn under the cable, creating another loop under the cable, before you start knitting.
And, by the way, because it's easy to inadvertently make an extra stitch or drop a provisionally cast-on stitch using this method, you might discover at the end of your first round that your stitch count is off by one or two. Just increase or decrease stitches to compensate--it's likely that no one will be able to detect the mistake.
As noted earlier, my preference is for the waste yarn method. Having waste yarn securing the first row of stitches also ensures that your stitch count will be accurate after the foundation round is worked. However, the wasteless method is handy when you don't want to mess with waste yarn.
There are other variants: for example, you could work a loose backwards loop cast-on onto a circular needle first, push the cast-on stitches back off the needle and onto the cable, and then coil the needle around and thread the other needle end through the bottom of the cast-on stitches. This avoids the invisible provisional cast-on maneuvering, but doesn't really present an advantage: the invisible cast-on is not difficult, and it's harder than you think to thread the needle through the bottoms of those stitches. (It might be a little easier if you use an interchangeable needle set, and cast on using a larger needle point, then swap needle heads for the proper size.)