A true Mobius strip doesn't have "sides" like the paper model of one does. You can see that in the quote from Martin Gardner where he talks about a creature completing a loop becoming a mirror image with respect to it's friends. I assume the author wanted the paper model to mimic that property.

Incidentally, the author's choice makes for a much more interesting puzzle than the one you would have them use (IMO)--yours would be indistinguishable from a puzzle written on a cylinder.