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.