Claims about human behavior or cognition face challenges given the cultural diversity of the human species. Invariance testing and item bias analyses have been proposed to address questions about the comparability of data, but several recent reviews show that these tools are not widely used. Furthermore, lack of invariance in large scale studies led to increasingly vocal criticisms against the concept of invariance testing for culture-comparative research. We contribute to recent discussions about the role of invariance testing by highlighting that these debates confuse different questions and fail to consider the larger picture of empirical research within science. We argue that invariance testing is partly a theory-testing process and not only a purely methodological question. We distinguish five mapping processes that occur during a typical research project: 1) mapping ideas to theoretical constructs and concepts, 2) mapping constructs to stimuli, 3) participants` responses to stimuli are mapped onto numerical representations, 4) multiple responses to stimuli are tested for their internal relations (the typical focus of statistical invariance testing) and 5) empirical observations are mapped back to theoretical statements. Much of the literature has focused on the 4th mapping, but implications are relevant for theory-building beyond purely statistical criteria. We provide examples of these different mappings and how researchers can strengthen their theoretical and methodological toolkit in culture-comparative work. Our plea is that researchers pay more attention to invariance testing as a theory-guided process, because it offers important nuances about instruments, construct validity and psychological theories that are currently missed by focusing on only the statistical details. Claims about psychology valid for all humans depend on questions of invariance in the broad sense that we outlined here.