This should clear things up a bit!!!
Harmonic Reciprocal Mean Curvature Surfaces which are theta-isothermic
What is "theta-isothermic"? As one can define isothermic just by viewing on the functions of the fundamental forms - the Hopf differential is real with respect to a proper conformal coordinate - we define a surface to be theta-isothermic if the imaginary part of the Hopf differential is constant (theta) with respect to a proper conformal coordinate. However, neither is theta an invariant of the surface nor is the constant of the imaginary part of the Hopf differential a geomertrical property that is really understood up to now.
On the other side - even this is very new and not yet understood - for these "generalization" of isothermic surfaces there exist an involution, that maps a theta-isothermic surface (up to a scaling of the ambiente space) into an isothermic surface in either S3 or H3 and vice versa.
thanks, but I haven't had enough to drink to understand that yet