First off, the top of the book is always turning toward the camera. There is no reversal of the angular velocity vector here.
Secondly, this is a book, an object with three very distinct principal axes. Labeling the principal moments of inertia as A, B, and C, with A<B<C, for this book the ratio of the largest moment of inertia to the smallest, C/A, is about 3. The intermediate unstable axis B has an moment of inertia that is just about ideally placed in terms of maximizing tumble. Compare that to the Earth. The Earth is very close to a symmetric top; the ratio (B-A)/C is very, very tiny. Moreover, the ratio (C-A)/C is about 1/309. The large scale tumbling seen in that video becomes a tiny little thing called the Chandler wobble with the Earth.
Thirdly, this is a book, a rigid body. The Earth is an elasto-plastic body. This makes the behavior deviate from that of a rigid body. The Chandler wobble doesn't have quite the frequency one would expect for a rigid symmetric top. The magnitude of the wobble oscillates, alternately damped and excited by the polar tide.
There is a phenomenon called polar wander, but this does not involve the Earth spinning in the opposite direction. Apparent polar wander results from the motion of the tectonic plates. True polar wander results when the mantle changes orientation. The Earth's rotation axis when viewed from an inertial frame doesn't wander. It is the continents, and possibly the mantle, that wander. This is a very slow process, a degree or so per million years. Whether true polar wander ever did occur remains a bit contentious. There are several articles in the scientific literature arguing for various true polar wander events. There are also articles arguing against such events.