It is well-known that bijection between $\textbf{R}^p$ and $\textbf{R}$ exist (e.g. here, though many other examples exist).

The problem with all these examples of bijections is that typically the mapping does not have any structure (obviously, it cannot be continuous); and most examples do not even make an attempt at providing an $f$ that has some structure.

My question is: What bijections $f:\textbf{R}^p \to \textbf{R}^q$, where you are allowed to specify natural numbers $p\neq q$, do you know that are "nice"?
"Nice" can mean that some basic geometric property is preserved, it can mean that $f$ is interpretable in some way, or it can mean that it is in some other way appealing.