Never heard of such a thing. Don't see how it could be possible unless the collet is turning separately from the spindle. This should be impossible because the spindle has a pin that must fit in the collet's keyway in order to be tightened. You must be missing the pin or it has been buggered up to the point it is no longer long enough to fit in the keyway.
EDIT: I assumed you were running 16C or 20C collets. Not familiar with that model. We are running 6 Hardinge barfeeds. Got the first one about 20 years ago. None are Twin spindles, tho. Did you order it with different collets? Or does it come with different collets than the two I mentioned? We have a Nakamura TW20 that uses B42 collets. There is no keyway on these. However, we have never experienced the problem you describe. No keyway sure makes it hard to automatically load a new bar of hex material.
