I'm confident with rotations as 2D matrix and used to manipulate coordinates in a orthonormal plane.
But I'm very confused when this plane has $Y$ inverted (use case: screen or paper with a top left origin).
What is the direction of rotation when I give a positive $Theta$ rotation in this inverted basis?
The conventional representation of a Cartesian plane with a horizontal $x$ axis pointing to the right and a vertical $y$ axis pointing upward gives us a counter-clockwise rotation for a positive rotation angle.
But the rotation formulas just deal with numbers, not with drawings. They are all about what coordinates you put in and what coordinates come out, not where you draw those coordinates on the paper or on a display screen.
For example, the rotation by $\theta = \frac\pi2$ takes points on the positive $x$ axis to points on the positive $y$ axis, and in the meantime takes points on the positive $y$ axis to points on the negative $x$ axis. For example, it takes $(1,0)$ to $(0,1)$ and takes $(0,1)$ to $(-1,0).$
With the usual "$x$ right, $y$ up" orientation of the axes, that's a counterclockwise rotation. With the $x$ axis pointing right but the $y$ axis pointing down, it's a clockwise rotation.