Spinning wheel on rotating stool - why am I also affected even when the wheel axis is horizontal? So I can point the axis of spin in one horizontal direction - and then move it in another direction (so it stays horizontal - like the picture below). And that still changes the direction of spin (on the rotating stool). Why does that happen? 
I already know the full explanation in the case that the wheel axis is vertical and the wheel is flipped over.

 A: Let me set up the following convention for orientations, loosely based on aviation.
I take the bicycle wheel as reference.
Swivel: rotation around vertical axis
Roll: like roll of an airplane
Pitch: like pitching of an airplane.
Let the spinning of the bicycle wheel be designated as 'rolling'.
InquilineKea, in the picture I presume the bicycle wheel rolling, and I gather the bicycle wheel is swiveled, while preventing pitch of the bicycle wheel.
I gather you observe that as the bicycle wheel is swiveled the chair, with the bicycle-wheel-holding-person in it, tend to counter-swivel. (I'm assuming the entire weight of the bicycle wheel is supported by the person in the chair.)
Discussion:
As the bicycle wheel is swiveled you clearly need to prevent it from pitching. The torque to counter-act pitching tendency is a pitching torque. Clearly this pitching torque will not affect swivel of the chair.
So I concur with you: it seems to me this excludes the spin of the bicycle wheel from the explanation.
Next step:
Try the same with non-spinning bicycle wheel. Swiveling the bicycle wheel will affect the chair at least somewhat, as the bicycle wheel has a considerable moment of inertia.
It would take a very strong person indeed to prevent any pitching of the bicycle wheel; the motion will be somewhat jerky. Possibly that jerkiness is muddling the effect.

