An operator of belief change is reconstructible as another such operator if and only if any outcome that can be obtained with the former can also be obtained with the latter. Two operators are mutually reconstructible if they generate exactly the same set of outcomes. The relations of reconstructibility among fifteen operators of contraction, including the common AGM contraction operators, are completely characterized. Furthermore, the additional such relations are characterized that arise if all belief sets are required to be finite-based (...) or if the language is finite. These results are significant for the choice of a formal model to represent how real-world agents change their beliefs. In particular, if two contraction operators are mutually reconstructible, then that limits the types of considerations that can legitimately be invoked in favour of one over the other. (shrink)