I really wanted to stay within the 2 sq in, or 13 sq cm constraint.  Not just for cost, but for the challenge.  I want the experience of laying out a dense circuit in Kicad.

Initially I assumed I could do a central hub with long arms sticking out that would be joined with solder once the components are installed.  When measuring just the solid area of my designs, I could create a much larger mobius strip.  It turns out that OshPark (and I suspect most vendors) measure your design by it's bounding box, or the smallest rectangle it will fit into.

With this in mind I investigated layouts that could be cut apart, then re-assembled in a different configuration.  I started looking at the minimum area needed for a central hub containing the microcontroller.  Measuring an Atmega 328p QFP I had on hand, I found that this chip has a 12mm x 12mm footprint.  This seemed rather large compared to 2 sq inches, so I decided to use the lead-less VQFN version of the chip.  It measures only 5x5mm, which allowed me to fit my central hub including current limiting resistors and interconnects into 16mm x 16mm.  At this point I also decided to use 0603 components instead of my go-to and easy-to-place 0805s.

Placing this at one end of a 16mm x 81.25mm rectangle leaves enough material for two additional strips, 8mm x 65.25mm.  These strips will contain the LEDs on both sides, and will be soldered together, and to the hub, with the mobius twist.