On an $m\times n$ grid there's a snail in each cell. Each round, the snail army can choose four snail and make them perform the complete Quadrilateral Dance, which is rotating the four snails clockwise by $90$ degrees, moving each one to an adjacent cell. Find all pairs of positive integers $(m,n)$ such that the snails can achieve any permutation by performing a finite number of times of Quadrilateral Dance. Proposed by chengbilly