[lupire]

It maybe more clear if you visualize the reflection as a pair of perpendicular reflections, first across the diameter (which is also across the center) and then internally reflecting the diameter (which is again also across the center.)

Two reflections with a common fixed point make a rotation around that fixed point (angle of reflection is double the angle between the reflection axes.). Two perpendicular reflections make a 180 degree rotation around the intersection of the axes of rotation.