The trick is that the cat can create and destroy angular inertia at will by moving its legs towards or away from the axis of rotation. Think of the front and back of the cat as being separate rotational systems with the same axis. The first thing the cat does is
increase the angular inertia of the back half and
decrease the angular inertia of the front half. Now the system is unbalanced. The cat twists the front half around while the back half, with a much higher angular inertia, moves very little. Next, the cat reverses the roles of front and back by decreasing the angular inertia of the back half and increasing the angular momentum of the front half. Now the back half moves rotates all the way around while the front half moves very little.
A linear analogy would be a two balls connected by a spring oscillating in a vacuum. If one ball is much lighter than the other -- like a football connected to a cannonball -- it moves a lot, while the cannonball moves very little. Let's say the football and the cannonball are on a football field (a frictionless, vacuum-filled football field, of course.) The cannonball is on the 50 yard line, staying roughly stationary, and the football is oscillating back and forth between the 49 yard line and the 47 yard line. The system can't go anywhere if it starts with zero linear momentum, because the mass (linear inertia) of the balls remains fixed. But imagine you can change the masses of the balls at will. When the football reaches the 47 yard line, you swap the masses of the balls, so the football becomes very heavy and the cannonball becomes very light. Now the football stays almost stationary at the 47 yard line, and the cannonball oscillates between the 48 yard line and the 50 yard line. When the cannonball reaches the 48 yard line, you swap the masses so that the cannonball is once again very heavy and the football is very light. Now the football is oscillating between the 47 yard line and the 45 yard line. By swapping the masses repeatedly you can move down the field all the way to the end zone.
Because of conservation of mass/energy, you can't actually do that trick with linear inertia, but you can do it with angular inertia. The front and back half of the cat are like the football and the cannonball, and the cat controls the angular inertia of each half by tucking and extending its legs. When the cat is upside down, it increases the angular of inertia of its back half by extending its back legs, and it decreases the inertia of the front half by tucking its front legs. Its back half acts like the cannonball, and its front half acts like the football. That allows it to twist its front half around while the back half stays almost stationary. Once the front half is right side up, it extends its front legs and tucks its back legs, effectively swapping the football and the cannonball. In zero gravity, a cat could twist itself around that way as many times as it wanted: just like the football and the cannonball working their way down the football field, except that it wouldn't violate any physical laws.