= Cross-ratio permutation orbit
{title2=$\{\lambda,1/\lambda,1-\lambda,1/(1-\lambda),(\lambda-1)/\lambda,\lambda/(\lambda-1)\}$}
For the convention in <cross-ratio normalization of four projective points>, permuting four distinct <projective points> produces the six displayed rational expressions. Interchanging the first pair sends $\lambda$ to $1/\lambda$; interchanging the middle pair sends it to $1-\lambda$. The four permutations consisting of the identity and the three double transpositions form a <Klein four-group> preserving the <cross-ratio>. The resulting six-element action gives every displayed expression.
For real $\lambda\notin\{0,1\}$, repeated values occur exactly for $\lambda\in\{-1,1/2,2\}$, and then the <orbit> has size three. To prove this, a repetition gives a nonidentity stabilizer after composing one operation with the other's inverse. Its fixed-point equations give $-1$, $1/2$, $2$, or $\lambda^2-\lambda+1=0$, with the last case nonreal. With one point at infinity, $\lambda=(x_1-x_3)/(x_1-x_2)$; the three real exceptional values mean that one finite point is the midpoint of the other two. Thus these <harmonic conjugate points> configurations have three collinear equally spaced finite points in that affine coordinate.
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