What do you mean by symmetric change? In other words, if a stone is at (x,y) then how does this map to (x',y') upon a symmetric change?
We’ve got 4 stones and we put them on the plane in points (0,0), (0,1), (1,0) and (1,1). In one move we choose one stone and we change his position symmetrically under one of another stones. Is it possible that after finite discontinuity numbers of moves, three stones belong to one straight?