If an ant crawled along the length of a mobius strip, it would return to its start after crawling the entire length of both sides, never crossing an edge. If you draw a line down the center of a moebius strip, the end of the line will meet the start of the line and you will never have to pick up your pencil to make that happen. And if you cut the strip along that center line, you end up with a continuous strip with two twists in it.
Elizabeth Zimmermann understood the design potential of the moebius. This is her “Moebius Ring” pattern from Schoolhouse Press Wool Gathering #28. It’s knitted in garter stitch with an I-Cord border. You start with a provisional cast on. Knit forever. Then give it one twist. Mathematicians know a moebius can have a clockwise twist or a counterclockwise twist, but that matters not a bit when it comes to a scarf. Then graft your end row to your beginning row and the result is a very sensible long cowl, scarf, or combination hood and scarf.