  We use geometric arguments to prove explicit bounds on the mean oscillation
for two important rearrangements on $\mathbb{R}^n$. For the decreasing
rearrangement $f^*$ of a rearrangeable function $f$ of bounded mean oscillation
(BMO) on cubes, we improve a classical inequality of Bennett--DeVore--Sharpley,
$\|f^*\|_{BMO(\mathbb{R}_+)}\leq C_n \|f\|_{BMO(\mathbb{R}^n)}$, by showing the
growth of $C_n$ in the dimension $n$ is not exponential but at most of the
order of $\sqrt{n}$. This is achieved by comparing cubes to a family of
rectangles for which one can prove a dimension-free Calder\'{o}n--Zygmund
decomposition. By comparing cubes to a family of polar rectangles, we provide a
first proof that an analogous inequality holds for the symmetric decreasing
rearrangement, $Sf$.
