Standardizing aspects of assessments has long been recognized as a tactic to help make evaluations of examinees fair. It reduces variation in irrelevant aspects of testing procedures that could advantage some examinees and disadvantage others. However, recent attention to making assessment accessible to a more diverse population of students highlights situations in which making tests identical for all examinees can make a testing procedure less fair: Equivalent surface conditions may not provide equivalent evidence about examinees. Although testing accommodations are by now standard practice in most large-scale testing programmes, for the most part these practices lie outside formal educational measurement theory. This article builds on recent research in universal design for learning (UDL), assessment design, and psychometrics to lay out the rationale for inference that is conditional on matching examinees with principled variations of an assessment so as to reduce construct-irrelevant demands. The present focus is assessment for special populations, but it is argued that the principles apply more broadly.