I was playing around in Desmos and noticed that the plots of y=x^2 and x=y^2 are 90 degree rotations of each other. This also appears to work for y=cos(x) and x=cos(y) as well as y=floor(x) and x=floor(y). However it is not true for x=y and y=x obviously since those are the same, and on inspection not for y=ln(x) and x=ln(y).
Is there a rule of thumb for when this works and when it doesn't? Or is there some more general effect always occurring and the 90 degree rotations are a special case?
If you change the x for y it will always change, but it is not a 90 degree rotation, it is actually a reflection by the x=y line. This happens because you are not doing any operation on the equation, just changing the variables. A great way to think this explanation is trying to graph the same equations but this time the horizontal axis is going to be the y axis and the vertical axis is going to be the x axis.