In Conformal Tiling on a Torus, the author John M. Sullivan provides a conformal parameterization of the torus. The conformality is associated to the so-called aspect ratio $s = \sqrt{\frac{R^2}{r^2}-1}$. For example, the square torus is the one for which $s=1$, and that means that a square pattern conformally fits on this torus.
Can we "transport" this notion by a geometric inversion? The inversion of a torus with respect to a sphere is a cyclide. Does it make sense to talk about a "square cyclide" if the torus is square? And more generally does it make sense to talk about the aspect ratio of the cyclide?