Variables in many multivariate datasets are naturally organized into modules, giving rise to structured dependence within and between groups. This talk discusses covariance structure testing for modular data through the framework of commutative quadratic subspaces (CQS). We introduce representative covariance models and explain how their algebraic structure facilitates estimation and hypothesis testing. Likelihood ratio and Rao score methods are discussed in fixed-dimensional settings, followed by an overview of extensions and challenges in high-dimensional inference. Examples motivated by financial and biomedical data illustrate the interpretation of the models and their potential applications.