It is well known that the Earth spins on its axis. It is also well known that the Earth's axis also precesses, i.e. spins around a secondary axis, much more slowly. Less well known is that we have seen asteroids that seem to tumble through space, with precession rates on roughly the same order of magnitude as rotation rates. My question is, why haven't we seen many instances of higher orders of rotation (where the secondary axis spins around a tertiary axis, the tertiary axis spins around a quaternary axis, etc.)? Can rotations of a high enough order always be expressed in terms of lower-order rotations at higher rates of rotation, or do higher-order rotations eventually decay into lower-order ones (as, for example, rotating around the $x$, $y$, $z$, and $x$ axes, all at the same rate*, produces an about-face, a seemingly impossible behavior)? (The latter seems to be stated by a comment on this question, but it doesn't go into much explanation.)
*This generates the matrix $$\begin{pmatrix} \cos^2t & \sin t\cos t(\sin t-1) & \sin t(\sin t+\cos^2t) \\ \sin t(\sin t+\cos^2 t) & \cos t(\sin^3t-\sin^2t+\cos^2t) & \sin t\cos^2t(\sin t-2) \\ \sin t\cos t(\sin t-1) & \sin t(\sin^3t+2\cos^2t) & \cos t(\sin^3t-\sin^2t+\cos^2t) \\ \end{pmatrix}$$ with the about-face occurring at $t=\pi/2$.
