The set of polynomials under the action by a symmetric group

Maybe something along these lines?

rat = (x1^r x2^
      r (1 - x1 x3) (1 - x2 x3))/((1 - x2/x1) (1 - x3/x1) (1 - x3/x2));
vars = {x1, x2, x3};
subs = Map[Thread[vars -> #] &, Permutations[vars]]

(* Out[112]= {{x1 -> x1, x2 -> x2, x3 -> x3}, {x1 -> x1, x2 -> x3, 
  x3 -> x2}, {x1 -> x2, x2 -> x1, x3 -> x3}, {x1 -> x2, x2 -> x3, 
  x3 -> x1}, {x1 -> x3, x2 -> x1, x3 -> x2}, {x1 -> x3, x2 -> x2, 
  x3 -> x1}} *)

Now do the substitutions.

Map[rat /. # &, subs]

(* Out[109]= {(
 x1^r x2^r (1 - x1 x3) (1 - x2 x3))/((1 - x2/x1) (1 - x3/x1) (1 - x3/
    x2)), (x1^r (1 - x1 x2) x3^
  r (1 - x2 x3))/((1 - x2/x1) (1 - x2/x3) (1 - x3/x1)), (
 x1^r x2^r (1 - x1 x3) (1 - x2 x3))/((1 - x1/x2) (1 - x3/x1) (1 - x3/
    x2)), (x2^r (1 - x1 x2) x3^
  r (1 - x1 x3))/((1 - x1/x2) (1 - x1/x3) (1 - x3/x2)), (
 x1^r (1 - x1 x2) x3^
  r (1 - x2 x3))/((1 - x2/x1) (1 - x1/x3) (1 - x2/x3)), (
 x2^r (1 - x1 x2) x3^
  r (1 - x1 x3))/((1 - x1/x2) (1 - x1/x3) (1 - x2/x3))} *)