6. Let $\left(x_{1}, x_{2}, \cdots, x_{20}\right)$ be a permutation of $(1,2, \cdots, 20)$, and satisfy $\sum_{i=1}^{20}\left(\left|x_{i}-i\right|+\left|x_{i}+i\right|\right)=620$, then the number of such permutations is $\qquad$.
$$
\sum_{i=1}^{20}\left|x_{i}+i\right|=2 \sum_{i=1}^{20} i=420 \Rightarrow \sum_{i=1}^{20}\left|x_{i}-i\right|=200=2\left(\sum_{i=11}^{20} i-\sum_{i=1}^{10} i\right) \text {, }
$$

So the number of permutations that satisfy the condition is $A_{10}^{10} \cdot A_{10}^{10}=(10!)^{2}$.