Does the Oort cloud rotate? Or is it static?

Good question! Does the Oort cloud rotate?

Alright, to start with, it should be noted that while the Oort cloud is hypothesized to exist, its existence has not been confirmed.

The next question is, "Does the Cloud move?" To which the answer is most likely "yes," bearing in mind that there are two main forces at work in deep space:

Gravity refers to the force exerted by an object on another, including the pull of neighboring stars, the Sun, and other planets, as well as the elements of the Cloud on other elements.

the absence of friction to prevent the objects being pulled from moving.

The question then becomes, does the Oort Cloud revolve around the Sun (as in, does it revolve around the Sun or does it rotate on its own axis, i.e., the Earth spins and the Sun rises and sets, or the Earth revolves around the Sun and creates seasons)?

Once more, the answer is almost certainly yes. Remember that these objects aren't just floating off into deep space; rather, they are being pulled in a constant direction by the Sun's gravitational field. Since the objects aren't colliding with the Sun, the motion of the Cloud's objects, and consequently the Cloud itself, would orbit the Sun.

Using Saturn's rings as a more familiar example, the speed at which the rings revolve around the planet depends on its distance from the planet's surface.

NASA Space Math & Science Weekly 10Page 28.pdf

Because of its great distance and sparseness, the Oort Cloud is thought to be largely static, with individual comets orbiting at random times. As a whole, the cloud's rotation is not well defined.