For any $A\subseteq \mathbb{N}$ we let the (lower) density of $A$ be defined by $$d(A) = \lim\inf_{n\to\infty}\frac{|A\cap\{0,\ldots,n\}|}{n+1}.$$ If $\pi:\mathbb{N}\to\mathbb{N}$ is a permutation (bijection), let $\text{pos}(\pi) = \{n\in\mathbb{N}: \pi(n) > n\}$ and $\text{neg}(\pi)=\{n\in\mathbb{N}: \pi(n) < n\}$.

To me it seems inconceivable that there is a permutation $\pi:\mathbb{N}\to\mathbb{N}$ with $d\big(\text{pos}(\pi)\big) \neq d\big(\text{neg}(\pi)\big)$ -- but my intuition has let me down many times.

Is my intuition correct this time?