In this paper we report results for a reformulated random order approach to values of non-atomic games. The reformulation is achieved by generating random orders from a fixed subgroup of automorphisms Theta that admits an invariant probability measurable group structure. The resulting Theta-symmetric random order value operator is unique and satisfies all the axioms of a Theta-symmetric axiomatic value operator. For the uncountably large invariant probability measurable group ( Theta, B, Gamma) of Lebesgue measure preserving automorphisms constructed in Raut [1997], Theta-symmetric random order value exists for most games in BV and it coincides with the fully symmetric Aumann-Shapley axiomatic value on large subspaces of games in pNA. Thus by restricting the set of admissible orders and the space of games suitably, it is possible to circumvent the Aumann-Shapley Impossibility Principle for the random order approach to values of non-atomic games.