The picture below shows an isolated system with a fairly massive wheel at one end, attached via its axle to a long shaft, like a bike tire on a bike frame, but the bike frame is merely a low mass 'truss.' At the other end of this long shaft is a mass of roughly the same magnitude as the wheel, so the center of gravity (CG) is roughly in the middle of the shaft. This mass is fixed to the shaft with no rotation possible. The assembly is floating motionless in 'space', with no contact to anything else.
If a motor on the 'truss' causes the wheel to spin in one direction, the truss would begin to spin in the other direction about the common center of mass. The net angular momentum would thus be zero both before and after the spinning commences. The wheel and the truss would rotate in the same plane. The actual rotation rate of each part is dependent upon the individual masses, wheel radius, and the length of the shaft. It is getting complex, but so far easy enough to envision in general.
Now, place into the 'truss' a mechanism that causes this long shaft to rotate on its long axis (which would be perpendicular to the spin axis) while it is still undergoing the above rotation relative to the wheel. Upon rotation at this new joint, a force is applied to the axle of the spinning wheel (and of course to the mass at the other end of the truss, too) causing each part to try to turn in the opposite direction. Given that the net angular momentum is still zero, the vector sum of the individual parts (the spinning wheel and the body as a whole rotating around its CG) must be equal and opposite. I cannot envision the result, however, nor do I have the background to derive the equations of motion of this 'whirly gig.' The best I can imagine is that the plane of overall rotation should change, with the plane of the wheel's rotation changing in the opposite direction. This would seem to result in a state where the wheel is no longer spinning in the same plane as the overall assembly, but that would seem unsustainable and perhaps unstable. How can the behavior of this system be understood?