Pivot point is not tied to any element, it’s the setting in the pivot menu like mentioned.
But pivot point is only part of the story when doing rotations because you usually want to do it along some axis. That’s where the next menu on the header comes in. If you want to do rotation along some other axis that is aligned with one of the Global axes, you need to change the transform orientation to get the axes aligned. It’s also possible to make a custom transform orientation by selecting a mesh element and pressing ctrl+alt+space.
For example, you have a face rotated to some orientation and you want to rotate it 90° along one of its edges. Couple of ways to do that. You could:
- switch to active element pivot (alt+period), switch TO to ‘normal’ (alt+space), select the face, change select mode to edge (ctrl+tab), and then select the edge last, press R to rotate, then axis key twice (xx/yy/zz) and then type in 90 (and - to switch direction). The order you input axis, amount and direction doesn’t matter.
- select the edge, snap the 3d cursor to it with shift+S -> cursor to selected, make a custom TO with ctrl+alt+space, switch pivot point to 3d cursor (period) and then again, R,xx/yy/zz, 90.