This phenomenon is one that always feels surprising. It's odd enough when you can see that you're holding a spinning wheel, but there's a practical joke where you hand someone a suitcase containing a spinning wheel, and they really can't figure out what's happening.
The key point is that the 'motion' that's being resisted is not simple acceleration in a straight line. That feels completely normal. It's twists that feel strange. That's because the wheel has angular momentum around its axis, and when you try to make the axis point a different way you're changing the direction of the angular momentum. That requires a torque. If the wheel wasn't spinning, you wouldn't need that torque. You can read all about this, with nice pictures, in any standard beginning mechanics book. If you search the Web, maybe try 'vector angular momentum'.
The basic feel of the whole experiment should be just the same in space. The only thing gravity has to do with it is that if the wheel is supported on one end of the axle, gravity will exert a torque around that point. That's what makes the wheel 'precess' around. The way it feels to twist the wheel doesn't depend on that.
As far as the reference frames go, it is certainly easiest to use a Newtonian or Special Relativistic frame, in which the twist reflects a change in the orientation with respect to the prior orientation. I think if one considers more general rotating frames, there are pseudo-forces inducing torques. So far as I know, Higgs has nothing to do with it.
The contribution of the spinning to the mass is quite small, since the velocities in the average rest frame of the wheel remain very small compared to the speed of light.
(republished on 07/13/06)