This isn't in any way a complete answer but I do find it illuminating to
see this behaviour in a numerical simulation. In particular we see the  periodic flips that seem to appear out of nowhere. Note how angular momentum is temporarily stored in the $\omega_1$ and $\omega_3$ components of the angular velocity during the flips.

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/B00Ws.png