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Divers actually start rotating their entire
bodies rather rapidly around a new (body-centered) axis. If you watch
the slow-motion replays of the olympic diving, it is clear that this
rotation can be started in the air.
Divers are taking advantage of the fact that the conservation of
angular momentum does not imply the conservation of angular velocity if
the body's moments of inertia change. As physicists, we are used to
this fact with respect to the magnitude of the angular velocity. It is
also true that the direction of angular velocity can change even while
angular momentum is conserved, if the moments of inertia change.
The moment of inertia I of a body is a tensor, not a scalar.
... If only the diagonal components are
treated, then each component of of angular velocity can change in
magnitude individually, then the x-component of angular velocity cannot
be turned into a y-component or vice versa.
When the off-diagonal components of
I are non-zero, then you can increase one component of
angular velocity at the expense of another.
You can always, of course, find a set of axes in which a constant I is
diagonal.