If you'll reread my message, you'll note that I claimed the parity
argument wasn't obvious to me even in the case of a qtw metric;
my reference to half-twists was intended along the lines of "this
is even less obvious". I'd be interested in seeing your "obvious"
proof in the qtw case, if you'd care to send it to me (no need to
bother the whole mailing list with it).

-- Don.