From the research I have seen it may be possible that a thin helmet designed to rotate relative to the wearers head may well be able to reduce the incidence of brain injury in cycling accidents. However, it could not presently be sold because it would not pass the current impact tests.
You can buy anti-rotation helmets. Passing the current impact tests is so easy you could get a crumpet to pass it.
I wouldn't bother with most of the anti-rotation designs because, like every other helmet, they fail at too low a velocity. Rotation from low speed impacts is only going to cause brain damage if you are freakishly unlucky or repeat them frequently, in the manner of NFL players.
It is also possible that a bobble hat could provide more protection against sudden rotational accelerations and decelerations of the head than current helmet designs.
Actually scalp movement and tearing provides built-in anti-rotation.
Some research, including the inspection of helmets after accidents, conclude that the foam was rarely, if ever, compressed as expected. This suggests that the dense foam currently used is too stiff and so not in fact protecting the brain.
That's a good question to ask... but in reality, no: most of the helmets recovered have cracked shells, and once the shell cracks then liner compression won't occur. That's why I'd recommend wearing a Snell B90 or 95 foam hat if you are going to bother wearing one - the tougher impact tests provide a much better guarantee of shell integrity.
This is born out by instances where the helmets of brain damaged victims are not damaged.
Brain damage will almost always be via rotation, so a current helmet will be irrelevant.
Only through a thorough understanding of what causes brain damage can effective solutions be identified. These solutions may in fact be very simple. Even a elastic net stretched over a current helmet would stop it from suddenly gripping the ground.
There is company that makes decent anti-rotation motorcycle helmets. They use a sliding membrane - it adds about 100Eu to the price. But for cycling helmets, worrying about rotation is pointless because they don't have the shell integrity to make it worthwhile.