What is Reddit's opinion of Explanatory Supplement to the Astronomical Almanac?

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Earth rotates (nearly) freely in space around a principal axis of rotation which, as you say, we know as the north-south axis. If you were to apply some torque to the Earth to change that, the rotation would change to some other axis. In other words, you wouldn't really see any more complicated behaviour than today, just a different axis, and there would be new "north" and "south" poles.

That's the boring answer. Now for something slightly more interesting.

Now as it happens the Earth does actually not rotate freely. It is subject primarily to tidal forces from the Moon and the Sun, but also a few others. This causes the axis of rotation to change over a period of roughly 26,000 years, an effect known as axial precession or precession of the equinoxes. The apparent north pole, currently in the vicinity of the star Polaris, will trace a circle over 26,000 years. The center point of that circle is the "orbital" north pole, i.e. the axis around which the Earth orbits (not rotates). This looks similar to how a spinning top sometimes slowly spins around its own axis of rotation, if you set it spinning with a slight tilt on a surface.

There are other changes to Earth's rotation too, they are called nutation, the principal effect of which varies over about an 18 year period.

So yes, it is possible to have such complicated rotation and in fact it already happens. However I do not wish to speculate on how exactly precession and nutation would look like if you modified the Earth's primary axis of rotation and/or angular momentum.

Source: <em>Explanatory Supplement to the Astronomical Almanac</em>.