Inequality involving rearrangement: $ \sum_{i=1}^n |x_i - y_{\sigma(i)}| \ge \sum_{i=1}^n |x_i - y_i|. $
We can prove it in a similar way as the rearangement inequality. There are only finitely many possibilities for $\sigma$, so a minimum is achieved, pick $\sigma$ so that it has the least possible number of inversions among all the permutations that minimize the expression.
Suppose by way of contradiction there is $i<j$ with $\sigma(i)>\sigma(j)$. Notice $|x_i-y_{\sigma_i}|+|x_j-y_{\sigma(j)}|\geq |x_i-y_{\sigma(j)}|+|x_j-y_{\sigma(i)}|$.
So the permutation that transposes $i$ and $j$ must also minimize the expression, and has less inversions, a contradiction.
For any convex function, such as $f(x) = |x|$,
$$ \sum f(x_i - y_{\sigma(i)}) \geq \sum f(x_i - y_i)$$
because $(x_i - y_{\sigma(i)})$ majorizes $(x_i - y_i)$.
A reference is the first theorem, 6.A.1, in chapter 6 of Olkin and Marshall's book on majorization, applied to the sequences $x_i$ and $-y_i$.
They attribute the result to a 1972 article by Peter W Day on general forms of the rearrangement inequality, and give a proof for vectors of real numbers. Day's article is about more general situations with ordered abelian groups. The inequality for real vectors must have been known earlier to many people.
You may be interested in the following inequality.
Let $f_1, \dots, f_n: \mathbb{R} \rightarrow \mathbb{R}$ be functions such that for all $1 \leq k < n$ the function $f_{k+1} - f_k$ is non-decreasing. In addition, let $y_1 \geq y_2 \geq \dots \geq y_n$. Then, $$ \sum_{k}f_k(y_{n-k+1}) \geq \sum_k f_k(y_{\sigma(k)}) \geq \sum_k f_k(y_k) $$
The book "the cauchy-schwarz master class" does not give a formal name for this inequality but just call it "a non-linear rearrangement inequality". See p.81 of the book.
I show a proof based on the inequality above.
Proof. Let $$ f_k(x) = |x_k - x| $$ Then $$ f_{k+1}(x) - f_k(x) = \begin{cases} x_{k+1}-x_{k} &\text{if}\ x \leq x_{k+1} \\ 2x - x_{k+1} - x_k &\text{if}\ x_{k+1} < x < x_k\\ x_{k} - x_{k+1} &\text{otherwise} \end{cases} $$ is non-decreasing, thus the inequality applies. That is, $$ \sum_k |x_k - y_k| \leq \sum_k |x_k - y_{\sigma(k)}| \leq \sum_k |x_k - y_{n-k+1}| $$