Intuitive way to understand the intermediate axis theorem: angular momentum is Iω but K.E. is ½Iω²; momentum is conserved but K.E. can dissipate; thus only max & min I are in equilibrium, and only max I (i.e. min K.E.) is long-term stable.
(This is not really rigorous, as it doesn't prove there aren't other stationary points of Iω̂, but it's close enough & helps to remember which axis is the stable one.)