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Many causal questions involve interactions between units, also known as interference, for example between individuals in households, students in schools, or firms in markets. In this paper we formalize the concept of a conditioning mechanism, which provides a framework for constructing valid and powerful randomization tests under general forms of interference. We describe our framework in the context of two-stage randomized designs and apply our approach to a randomized evaluation of an intervention targeting student absenteeism in the school district of Philadelphia. We show improvements over existing methods in terms of computational and statistical power.