Mobius transformation preserves the argument of the cross-ratio, which leads to the preservation of angles and generalized circles.
However, what about the preservation of the magnitude? Alternatively, what properties does the function $f(z)$ have?
$[f(z_1), f(z_2), f(z_3), f(z_4)] = k[z_1,z_2,z_3,z_4] \quad(k \neq 1)$
Edit: I suppose inversions are functions such that $[f(z_1), f(z_2), f(z_3), f(z_4)] = conjg[z_1,z_2,z_3,z_4]$