Is there a collineation which does not preserve betweenness?

Question

Consider the Euclidean plane $\mathbb{R}^2$. A collineation on $\mathbb{R}^2$ is a bijective function $f$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ such that the image of every line under $f$ is also a line. Does there exist a collineation which does not preserve betweenness? That is, there exists at least one triple of distinct points $(A,B,C)$ such that $B$ is between $A$ and $C$, but $f(B)$ is not between $f(A)$ and $f(C)$.