Chapter 2 describes five foundational principles of invariant measurement that define Rasch measurement theory (RMT) and that guide the development of explanatory Rasch models. These principles—(1) invariant unidimensional scale, (2) item-invariant measurement of persons, (3) non-crossing person response functions, (4) person-invariant calibration of items, and (5) non-crossing item response functions—provide the basis for objective and generalizable comparisons. The chapter explains how these principles ensure that item and person locations on a latent variable remain stable across different measurement conditions. Each principle is explored in relation to explanatory measurement models and demonstrates how covariates for items and persons can be incorporated into measurement models without violating the principles of invariance. Measurement should be independent of incidental factors, and should only reflecting the intended construct. Graphical tools such as Wright Maps illustrate these concepts by showing the simultaneous placement of items and persons on a continuum. The philosophical and statistical underpinnings of invariance are further explored. These ideas link to Rasch’s concept of specific objectivity and broader scientific ideals. By extending these principles with generalized linear mixed models (GLMMs), researchers can integrate measurement and explanation. This integration enables deeper insights into why items and persons vary in their locations on the latent scale. This conceptual framework sets the stage for the applied modeling in later chapters.