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By introducing the concept of “invariants”, Koffka (1935) endowed perceptual psychology with a flexible theoretical tool, which is suitable for representing vision situations in which a definite part of the stimulus pattern is relevant but not sufficient to determine a corresponding part of the perceived scene. He characterised his “invariance principle” as a principle conclusively breaking free from the “old constancy hypothesis”, which rigidly surmised point-to-point relations between stimulus and perceptual properties. In this paper, we explain the basic terms and assumptions implicit in Koffka’s concept, by representing them in a set-theoretic framework. Then, we highlight various aspects and implications of the concept in terms of answers to six separate questions: forms of invariants, heuristic paths to them, what is invariant in an invariant, roots of conditional indeterminacy, variability vs. indeterminacy, and overcoming of the indeterminacy. Lastly, we illustrate the lasting value and theoretical power of the concept, by showing that Koffka’s insights relating to it do occur in modern perceptual psychology and by highlighting its role in a model of perceptual transparency.