> if it actually wrapped around on itself with a half-twist, wouldn’t the letters wind up backwards too? Then 6-Down wouldn’t be MESSENGER but rather MESSENGER, and so on.
This premise is assuming the Mobius strip is transparent, which the header image clearly isn't. No letter-reversals or any symmetry is needed. On normal, opaque paper, you can just make a crossword as a normal long loop. (proof: Write GGGGGGG on one side of a paper, and write it upside-down on the other side. With a twist, you'll get one long GGGGGGGGGGGGGG... looping as if it were one boring single-sided loop).
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.