= Solution
The <cross-ratio permutation orbit> is generated by the two operations $\lambda\mapsto1/\lambda$ and $\lambda\mapsto1-\lambda$ from part (d). The full list is
$$
\boxed{\lambda,\quad\frac1\lambda,\quad1-\lambda,\quad
\frac1{1-\lambda},\quad\frac{\lambda-1}{\lambda},\quad
\frac{\lambda}{\lambda-1}.}
$$
These are all the possibilities: the four permutations generated by interchanging both pairs and by exchanging the two pairs leave the <cross-ratio> unchanged. They form the <Klein four-group> inside the <symmetric group> on four symbols, leaving at most $24/4=6$ different values; the displayed operations attain all six rational expressions.
There are fewer than six values exactly when a nonidentity operation in this six-element action fixes $\lambda$: if two operations give the same value, composing one with the inverse of the other gives such a fixed point. For real $\lambda\ne0,1$, the fixed-point equations are
$$
\begin{aligned}
\lambda=1/\lambda&\Longrightarrow\lambda=-1,\\
\lambda=1-\lambda&\Longrightarrow\lambda=\tfrac12,\\
\lambda=\lambda/(\lambda-1)&\Longrightarrow\lambda=2.
\end{aligned}
$$
The other two equations reduce to $\lambda^2-\lambda+1=0$, which has no real root. Hence
$$
\boxed{\lambda\in\{-1,\tfrac12,2\}\text{ are exactly the real exceptional values}.}
$$
Each exceptional value has the same three-element <orbit> $\{-1,\tfrac12,2\}$. Every other permitted real value gives six different values.
For the geometric interpretation, put $P_0=[1:0]=\infty$ and $P_i=[x_i:1]$ for $i>0$. Then
$$
\lambda=\frac{x_1-x_3}{x_1-x_2}.
$$
The three exceptional cases become
$$
\begin{aligned}
\lambda=-1&\Longleftrightarrow x_1=\tfrac12(x_2+x_3),\\
\lambda=\tfrac12&\Longleftrightarrow x_3=\tfrac12(x_1+x_2),\\
\lambda=2&\Longleftrightarrow x_2=\tfrac12(x_1+x_3).
\end{aligned}
$$
Thus \b[the three finite points are collinear and equally spaced, with one their midpoint], even when the $x_i$ are viewed as points of the <complex plane>. Equivalently they form an <arithmetic progression> in some order. These are the <harmonic conjugate points> configurations with the fourth point equal to the <point at infinity of the complex projective line>.
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