For example, how do I know that with:
$$f(x_1,x_2,x_3,x_4)=\frac{x_1 x_2+x_3 x_4-x_2 x_3-x_1 x_4}{x_1 x_2+x_3 x_4-x_1 x_3-x_2 x_4}$$
$f$ has the property:
$$f(x_1,x_2,x_3,x_4)=f(x_2,x_1,x_4,x_3)=f(x_4,x_3,x_2,x_1)=f(x_3,x_4,x_1,x_2)$$
I mean it's not that easy to discover all of them, right? I myself used to know only the first part of that symmetry until I find the completed one today. Is there any general method to examine the symmetry of a given function?