Imagine a ball (of uniform density) rolling on a horizontal plane without slipping. Due to its symmetry its angular speed in the reference frame of the plane will remain constant over time if it's not disturbed[1]. Its angular speed has three components. What I found surprising (but is obvious if we present it this way) is that by only nudging the ball horizontally you can't get it to spin around the vertical axis: so, horizontal nudges allow one to explore only a 2d subspace of the 3d space of the angular speeds.

tl;dr I got surprised by a consequence of the fact that while rotations around different axes do not commute, infinitisemal rotations (and thus angular speeds) do commute.

[1] If its density was not symmetric, it could undergo precession and nutation.