Can the sorting of a list be verified without comparing neighbors?
It is impossible. Suppose that you have the result of all comparisons except for the pair $(i,i+1)$. Then you wouldn't be able to distinguish between the following two cases:
$$
1,2,\ldots,i-1,i,i+1,i+2,\ldots,n \\
1,2,\ldots,i-1,i+1,i,i+2,\ldots,n
$$