In 2001: A Space Odyssey, Dr. Frank Poole a set adrift by his EVA pod that is controlled by HAL. He is doomed to death, just spinning in space. (Bowman rescued him a few minutes later.)
So it got me wondering… if I am floating around in space and spinning, is there anything I can do to stop the spinning? Would flailing my arms and legs in the opposite rotational direction stop the spinning?
Without friction from air resistance, no. You would need to throw away mass in the opposite direction of your spin.
The problem with moving your arms and legs one way is that, although it may slow you down temporarily, your arms are attached, and so have to stop moving pretty soon, so you’ll just keep rotatingh. It won’t stop you.
The problem is that a spinning person in space has angular momentum, the tendency of a body to keep rotating. unless you can apply an external torque, you;ll keep rotating. External jets, facing into the direction of rotation, can apply this torque.
Without external torque, you can’t change your angular momentum. You can alter your rate of spin by changing your moment of inertia (a measure of how your mass is distributed relative to your spin axis). Draw your arms in, and you’ll spin faster, Extend them and you’;ll spin slower. But you’ll still spin. As Heinlein liked to say, these tricks enable a figure skater to pull off some of their flashier tricks.
If you can set something on you spinning in the same direction, but separate from yourself, you can remove the spin from you. Part of the initial system will still be rotating (you can’t lose that angular momentum without torque), but at least you won’t be spinning anymore. So let’s say youy’re wearing a heavy belt, like a diver’s belt (for bo good reason, but work with me). You loosen it so that it forms a big loop around you that will clear your body, like your own private Saturn ring system. Start it rotating without pushing it physically away from your – just pushed sideways, in the direction of rotation. The belt speeds up, and you slow down. It’ll feel as if you’re pushing against a semi-fixed ring to slow yourself down, which you are. If you’re careful, you can get the belt spinning (faster than you were initially) and yourself not spinning at all. You haven’t stopped the angular momentum, but at least you got rid of any that your body (and spacesuit) had initially, and given it all to your belt.
Don’t touch the belt, or you’ll undo everything you’ve done.
Rotating your arms in the opposite direction would slow your bodies rotation, that is until you stopped rotating your arms, at which point your spinning would come back to its initial rate.
Conservation of angular momentum. You body as a whole will retain its angular momentum, although you can act to redistribute it, you can’t change it. Not without interacting with another mass.
You could carefully spin something up, so that they carry the momentum. In principle you could use your air supply as a reaction mass. Or your helmet. This is a self limiting technique.
Or you could catch something passing by at the appropriate point. You will gain linear momentum too, but catch it off your own c of c and you will spin or despin.
Or bounce something of a nearby large object.
Or just happen to carry a large magnet and try to interact with a local field.
Or you could define yourself as the reference and complain that the rest of the universe was spinning.
Ninja’d on a few of these 
Superb explanations both.
Just a quick add that this bit of physics is used in modern satellite design via a device called a reaction wheel. Reaction wheel - Wikipedia
I think you would do better to move the belt as a point source as far away from you as possible and then push off it. If you pushed off fast enough, neither you nor the belt would be spinning in the conventional sense. You would just be moving in opposite directions, but the angular momentum of the combined system would still be conserved. Think of it like the iceskater with one arm holding on to the flying belt and the other holding you gradually extending her arms out to infinity.
If you did that and got the push wrong, you’d be left wiuthout the belt and no way to fix your spin. If you have the belt spinning around you then you can fine-tune your spin and get yourself stopped.
Another thing is that pushing off from the belt gives you linear m,omentum as well, which may or may not be good, givedn the circumstances. With the belt around you, you stay in the same place.
Just poke a little hole in the outermost point of your spacesuit that is pitching toward the rotation angle.
The escaping air from your suit should offer up some counter rotation braking.
You would however have a new problem at that point.
On the other hand, using the belt that way makes it really easy to get it right (or at least, close enough to right), if the belt is long enough.
How so?
If you happen to have a powerful electromagnet on you, you could use it as a magnetic torquer. You are pushing against the earth through its magnetic field, transferring angular momentum from you to the earth.
Nitpick: magnetic torquers can only generate force perpendicular to the local magnetic field gradient.
So depending on your spin orientation versus the local magnetic field a magnetic torquer could be a perfect solution, no help at all, or (most likely) would help reduce your spin in one axis while increasing it in another.
But it is a neato invention. And the kind of thing you could imagine a Spaceman MacGuyver whipping up in a crisis.
You should be able to change your attitude of rotation - that is, if you are tumbling head over heels, you shoukd be able, without changing your total angular momentum, to turn so that you are rolling instead.
I suspect Chronos means this method: Hold the belt by one end (the lighter end if there is one). Your rotation will result in the belt extending itself into a straight line, increasing your total moment of inertia, and thus slowing your rotation substantially. Simply let go of the belt whenever you like: your rotation rate will remain small, as the belt goes off with a significant portion of your original angular momentum, with no need for precise aiming or timing skills. The longer and heavier the belt the better, of course.
Right, and that method is actually used sometimes for spacefaring payloads. I first heard of it for sounding rocket missions, where you’ve got only a very short time to shed any spin that you might have, and need to get it right quickly.
I think you’re referring to yo-yo de-spin systems? The idea is to put small weights on long wires, let out the wires and then let go. This allows small weights to carry away a lot of angular momentum. They are used on sounding rockets because those sounding rockets are spin-stabilized; they are also used on spin-stabilized upper stages on satellite launchers.
