We consider exchange markets with heterogeneous indivisible goods. We are interested in exchange rules that are efficient and immune to manipulations via endowments ( either with respect to hiding or destroying part of the endowment or transferring part of the endowment to another trader). We consider three manipulability axioms: hiding-proofness, destruction-proofness, and transfer-proofness. We prove that no rule satisfying efficiency and hiding-proofness ( which together imply individual rationality) exists. For two agents with separable and responsive preferences, we show that efficient, individually rational, and destruction- proof rules exist. However, for some profiles of separable preferences, no rule is efficient, individually rational, and destruction- proof. In the case of transfer-proofness the compatibility with efficiency and individual rationality for the two-agent case extends to the unrestricted domain. If there are more than two agents, for some profiles of separable preferences, no rule is efficient, individually rational, and transfer-proof.