It may be helpful to do some simple experiments, for example:
Try dropping a broomstick onto a tile floor. Observe what
happens.
For starters, for simplicity, let the initial rotational
rate be zero; that is, hold the stick at some angle theta
relative to the vertical, and just drop it.
Some specific points to ponder include:
-- Let's label the ends of the stick A and B. Does the
stick pogo on the B end multiple times, or does it clatter,
hitting one end and then the other, A,B,A,B.... ?
The special cases of theta=0 and theta=pi/2 are easily
understood; we exclude them from further consideration.
-- Let point X designate the point of _first_ impact.
Without loss of generality, suppose the stick initially
leaned to the right. Does the end of the stick end up
to the right of point X, or to the left of point X?
Note that this is about the simplest irregular body you can
imagine. The body is essentially one-dimensional, i.e. only
slightly more elaborate than a point particle, but its motion
is quite a bit more elaborate.
Two- and three-dimensional bodies are even more interesting.