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Interesting question.
Don't forget all the axes will be moving in one way or another (either due to their precession eg for a gyroscope or as a result of the motion of whatever they're attached to eg human, car, Earth).
So, in a simple case of there only being one spinny thing on the entire planet (and we'll assume the rest of the Universe including the planet is entirely stationary for now) then as I see it there are the following states:
1) axis is stationary and the spinny thing is simply rotating about it. The end points of the axis extend to infinity in both directions. From the point of view of standing outside the Universe (imagine it's inside a crystal sphere for now) the end points appear as antipodal dots on the Universal sphere.
2) axis is tumbling precisely end over end. Each of the ends of the extended axis describes its own "great circle" on the Universal sphere, and these circles overlay each other perfectly (though they're being drawn half a cycle out of phase with each other).
3) axis is precessing. Each of the ends of the extended axis describes a "small circle" on the Universal sphere, and these circles do not overlay one another at all.
4) axis movement is chaotic. The pattern made on the Universal sphere by the extended ends is also chaotic.
Skipping past all the rest of the calculations for now, you'll have to hold a standard football 1.32 ly from a stationary earth, or 1 x 10^49 ly from a moving one. Don't forget to multiply by the number of spinny things, and *always* check the source.
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simon
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