When things turn or spin on an axis, like the seats on a whirling amusement park ride, they gyrate. When you spin a top, you watch it gyrate.
An axis is a fixed, imaginary point or line that something rotates around — the Earth, famously, turns on an axis. If an object spins rapidly around this point, it gyrates. Many kinds of engines gyrate, and so do pinwheels, windmills, and gyroscopes. Even Elvis's hips were said to gyrate! The root of the verb gyrate is the Latin word for "circle," gyrus, which in turn comes from the Greek gyros, "circle or ring."