Abstract In this paper we study the exchange of indivisible objects where agents’ possible preferences over the objects are strict and share a common structure among all of them, which represents a certain level of asymmetry among objects. A typical example of such an exchange model is a re-scheduling of tasks over several processors, since all task owners are naturally assumed to prefer that their tasks are assigned to fast processors rather than slow ones. We focus on designing exchange rules (a.k.a. mechanisms) that simultaneously satisfy strategyproofness, individual rationality, and Pareto efficiency. We first provide a general impossibility result for agents’ preferences that are determined in an additive manner, and then show an existence of such an exchange rule for further restricted lexicographic preferences. We finally find that for the restricted case, a previously known equivalence between the single-valuedness of the strict core and the existence of such an exchange rule does not carry over.