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This paper is motivated by the open question whether the union of two disjoint NP-complete sets always is NP-complete. We discover that such unions retain much of the complexity of their single components. More precisely, they are complete with respect to more general reducibilities. Moreover, we approach the main question in a more general way: We analyze the scope of the complexity of unions of m-equivalent disjoint sets. Under the hypothesis that NE ≠ coNE, we construct degrees in NP where our main question has a positive answer, i.e., these degrees are closed under unions of disjoint sets.