Let θ be the orientation (angle) of a body (such as a cat), and let ω be its angular velocity.
It is well-known that θ can change even when the body is not rotating, using the conservation of angular momentum; that is, even when ω = dθ/dt = 0. That's how cats land on their feet so well.
But how can θ possibly ever change, when its derivative is zero?! What's wrong with the math?