My geometric intuition has failed to tell me that there are different sizes and shapes of Dedekind cuts. I realized it in the course of writing this answer only by doing algebra.

If we define a Dedekind cut for present purposes as a set $(-\infty,\alpha)\cap\mathbb Q$ where $\alpha\in\mathbb R$ then $\mathbb Q\cap(-\infty,\alpha)$ and $\mathbb Q\cap(-\infty,\beta)$ are geometrically congruent Dedekind cuts precisely if $\alpha-\beta\in\mathbb Q$. The congruence is $x\mapsto x + \beta-\alpha$.

If $\alpha,\beta\not\in\mathbb Q$ and $\beta-\alpha\not\in\mathbb Q$ and $\beta-\dfrac\alpha2\in\mathbb Q$ then $x\mapsto \dfrac x 2 + \beta-\dfrac \alpha 2$ is a similarity but not a congruence between $(-\infty,\alpha)\cap\mathbb Q$ and $(-\infty,\beta)\cap\mathbb Q$.

Two Dedekind cuts have the same shape if there is a similarity between them and two of the same shape have the same size if there is a congruence between them.

Dedekind cuts of the form $(-\infty, \alpha) \cap \mathbb Q$ when $\alpha \in \mathbb Q$ are self-similar, i.e. similar to a rescaling of themselves.

Is all this just another way of talking about Diophantine approximations, or is there something else of interest to be said about this?