We introduce new axioms for the class of all TU-games with a fixed but arbitrary player set. These axioms require either invariance of an allocation rule or invariance of the payoff assigned by an allocation rule to a specified player in two related TU-games. Combinations of these new axioms are used to characterize the Shapley value, the Equal Division rule, and the Equal Surplus Division rule. The classical axioms of Efficiency, Anonymity, Equal treatment of equals, Additivity and Linearity are not used.