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Perhaps one of the confusions here (at least for me) is that looking at
Bruce's link (below) it is now clear that the claim is NOT that the
gymnast/diver imparts NO angular momentum on takeoff, only that they often
impart no angular momentum around the vertical axis--that is, they do impart
AM around a horizontal axis perpendicular to their direction of motion.
That this can be converted to AM around other axes without changing the
overall vector AM is what is apparently not well understood but seems to
involve techniques like the 'tilt twist'. These maneuvers would then be a
combination of imparted AM from the jump and reorientation of the body by
twisting different parts--just right.