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?
A rigid body can't change it's angle, but a cat is not rigid (it can move one part in one direction and other parts in the opposite direction, and effectively wiggle around the full circle).
Good answer from bobuhito. Here's another explanation. Satellites have reaction wheels (which are not gyroscopes) to help them change orientation.
If you sit still on a rotating stool, and you want to change direction, and you are holding a long heavy rod, simply hold the rod over your head and rotate it horizontally a couple times.
Your total angular momentum at all times is zero, but that's because there's a positive angular momentum in the rod, balanced by a negative one in your body. When you stop turning it, both you and the rod have changed direction.
