In this paper, we introduce the general causal cumulative model of ordinal response. The statistical part of this model is a new family of hierarchical or non-hierarchical generalized linear models that represent the distribution of the outcome as a thresholded latent distribution. Its defining feature is the new link functions, which are order-preserving in the sense that they allow for arbitrary effects in individual thresholds while preserving their order. We show how the model can be interpreted as a generalization of some Signal Detection Theory (SDT) models and some Item Response Theory models. We propose an approach to measurement of latent variables which seems to follow from the requirement that measurements should be interpreted in the context of a causal theory of the unobservable response process. In particular, we formulate a causal definition of measurement invariance that seems to match the examples of measurement bias found in the literature, and we show that the commonly accepted statistical definition is flawed; this leads to the recognition of the fundamental problem associated with item parameters and of the critical role of substantive theory and speculative reasoning. We illustrate how, by making use of the statistical properties of our model, this approach to measurement can be followed in practice, and we explain what kind of issues of a more technical nature it may help address and when. In particular, we identify the central causal assumption of SDT models, we show how this assumption can be tested, how introducing item parameters can lead to severely biased point and interval estimates of the target causal quantities, and how causal estimate bias can be accounted for by means of a causal sensitivity analysis.