The "moment of inertia" is actually a tensor. If a body is rotated about an axis that is not a symmetry axis (or, more generally, an axis for which the tensor is not diagonal), the angular momentum in not in the direction of the axis of rotation.
This is discussed in Mechanics terxtbooks.
I still don't really understand why not and my textbook doesn't supply much of an explanation.
Could someone please explain more thoroughly in terms a high school physics student might be privy to?