I am a little confused by your example, but the definition

  σf(x_1,...,x_n) = f(σx_1,...,σx_n)

is certainly consistent and I'm not sure what you were hoping to show with your example functions.

I'm really not sure what σ(2) is supposed to represent? σ acts on rational functions like f and g, so

  σf(5, 3) = f(3, 5) = -2

and

  σg(5,3) = g(3, 5) = 2

Of course σf and f might not be equal, but I don't see how that is a contradiction? Happy to try and clear things up.