Balancing networks are highly distributed data structures used to solve multiprocessor synchronization problems. Typically, balancing networks are accessed by tokens, and the distribution of the tokens on the network’s output specify the property of the network. However, tokens represent increment operations only, and tokens alone are not adequate for synchronization problems that require decrement operations. For such kinds of problems, antitokens have been used to represent decrement operations. It has been shown that several kinds of balancing networks which satisfy the step property, smoothing property, and the threshold property for tokens alone, preserve their properties even when antitokens are introduced. A fundamental question that was left open was to characterize all the properties of balancing networks which are preserved under the introduction of antitokens. In this work, we provide such a simple combinatorial characterization for all the properties which are preserved when antitokens are introduced.