We present some formal properties of (symmetrical) commutativity, the major criterion used in transactional systems, which allow us to fully understand its advantages and disadvantages. The main result is that commutativity is subject to the same limitation as compatibility for arbitrary objects. However, commutativity has also a number of attracting properties, one of which is related to recovery and, to our knowledge, has not been exploited in the literature. Advantages and disadvantages are illustrated on abstract data types of interest. We also show how limits of commutativity have been circumvented, which gives guidelines for doing so (or not!).