Example 6 If there exist integers $a_{i}, b_{i}(1 \leqslant i \leqslant \varphi(n))$, such that $a_{i}, b_{i}, a_{i}+b_{i}, a_{i}-b_{i}(1 \leqslant i \leqslant \varphi(n))$ all form a reduced residue system modulo $n$, find all possible values of the positive integer $n$.
When $(n, 6)=1$, take any reduced residue system modulo $n$, denoted as $a_{i}$.
Let $b_{i}=2 a_{i}$. Then
$$
a_{i}+b_{i}=3 a_{i}, a_{i}-b_{i}=-a_{i} \text {. }
$$

According to Property 2, they are all reduced residue systems modulo $n$.
Conversely, if $a_{i} 、 b_{i} 、 a_{i}+b_{i} 、 a_{i}-b_{i}(1 \leqslant i \leqslant \varphi(n))$ are all reduced residue systems modulo $n$, then there exists some $b_{k}=1$.
Consider $a_{k} 、 a_{k}+b_{k}\left(=a_{k}+1\right) 、 a_{k}-b_{k}\left(=a_{k}-1\right)$.
Among them, there must be an even number, and there must be a multiple of 3; but according to Property 1, these numbers are all coprime with $n$. Hence $(n, 6)=1$.
In summary, the positive integer $n$ satisfies the condition if and only if $(n, 6)=1$.