So If we reflect y=f(x) in x=a, we get the curve y=f(−x+2a) ?I'd call it a double reflection, first around x=pi/4 and then around y=1/2. Equivalently, and the way I described it initially, it is actually a rotation around the point (pi/4, 1/2). In the same way, reflecting in both y=0 and x=0 amounts to rotating by 180 degrees about the origin.
Replacing x with pi/2 - x reflects in x=pi/4, and replacing y with 1-y (that is, subtracting the result from 1) reflects in y=1/2. These are worth pondering until you see why!