  The Polya-Szeg\H{o} inequality in $\mathbb{R}^n$ states that, given a
non-negative function $f:\mathbb{R}^{n} \rightarrow \mathbb{R}_{}$, its
spherically symmetric decreasing rearrangement $f^*:\mathbb{R}^{n} \rightarrow
\mathbb{R}_{}$ is `smoother' in the sense of $\| \nabla f^*\|_{L^p} \leq \|
\nabla f\|_{L^p}$ for all $1 \leq p \leq \infty$. We study analogues on the
lattice grid graph $\mathbb{Z}^2$. The spiral rearrangement is known to satisfy
the Polya-Szeg\H{o} inequality for $p=1$, the Wang-Wang rearrangement satisfies
it for $p=\infty$ and no rearrangement can satisfy it for $p=2$. We develop a
robust approach to show that both these rearrangements satisfy the
Polya-Szeg\H{o} inequality up to a constant for all $1 \leq p \leq \infty$. In
particular, the Wang-Wang rearrangement satisfies $\| \nabla f^*\|_{L^p} \leq
2^{1/p} \| \nabla f\|_{L^p}$ for all $1 \leq p \leq \infty$. We also show the
existence of (many) rearrangements on $\mathbb{Z}^d$ such that $\| \nabla
f^*\|_{L^p} \leq c_d \cdot \| \nabla f\|_{L^p}$ for all $1 \leq p \leq \infty$.
