All the explanations given involve conservation of angular momentum, which is perfectly correct, but I feel that people with no thorough background in physics and mathematics will be left unsatisfied with this.

Is there a way to explain conservation of angular momentum in terms which would be understandable to the layman? It's a pedagogical problem I have given a lot of thought and I have yet to find a satisfactory answer.

Surely, you need to start from something, some basic axiom that the person will be willing to accept at face value. I thought about using either the law of action and reaction or conservation of momentum. I think these are relatively easy to describe "pictorially". But going from these to angular momentum using a vector product, is a mathematical procedure I'm not sure I can explain to someone who doesn't know anything about math. So, this should be circumvented in some way by a nice visual example again to make things clearer and I haven't found one. 

Anybody having ideas?

Why don't spinning tops fall over?
Is there a relation between the eigenvectors of a linear operator (Matrix) $A$ and eigenvectors of $A^TA$? This question is related to eigenvectors ans not eigenvalues. Further the size of the matrix $A$ and $A^TA$ are different if $A$ is not a square matrix. So my question has no relation to the question about eigenvalues of $A$ ans $A^T$.
