Which other way?

What you wrote is wrong. If (x,y) becomes (y,-x) (ie x is replaced with y and y is replaced with -x) it is NOT a 90 degree rotation anticlockwise. It is a 90 degree rotation clockwise.

If by the other way, you mean 90 degree anti clockwise, then place a point (x,y) in the first quadrant. Let the position of the x-value be smaller in comparison to the y-value. Now rotate this point 90 degree anti clockwise. Figure out what this point will be. Hint: The point will be (x,y), (x,-y), (-x,y), (-x,-y), (y,x), (y,-x), (-y,x) or (-y,-x).

Claimer: This method is not a proof but it does get the job done.