Again, is there a conformal self-map that interchanges two points in the upper half-plane? I'm beginning to think this isn't so. Such a map would be a FLT $\frac{az+b}{cz+d}, ad-bc=1$, with real coefficients. If I construct a map that sends some $w \mapsto t$ and $t \mapsto w$, I can't find values for the coefficients that work, though I may not have done this with sufficient generality.
I'm also thinking that given two points and considering a circle that they describe, the self-map would have to reverse the orientation of the circle which I don't think is possible.