We study the following question: Given two semidualizing complexes B and C over a commutative noetherian ring R, does the vanishing of Ext(R)(n) (B, C) for n >> 0 imply that B is C-reflexive? This question is a natural generalization of one studied by Avramov, Buchweitz, and Sega. We begin by providing conditions equivalent to B being C-reflexive, each of which is slightly stronger than the condition Ext(R)(n) (B, C) = 0 for all n >> 0. We introduce and investigate an equivalence relation approximate to on the set of isomorphism classes of semidualizing complexes. This relation is defined in terms of a natural action of the derived Picard group and is well-suited for the study of semidualizing complexes over nonlocal rings. We identify numerous alternate characterizations of this relation, each of which includes the condition Ext(R)(n) (B, C) = 0 for all n >> 0. Finally, we answer our original question in some special cases.
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