<p>This work introduces different approaches to model mean structures in matrix-variate normals, addressing the over-parameterization issue commonly encountered when exploiting these distributions in the context of model-based clustering. The methodology employs parsimonious parameterizations of the means using additive, interaction and/or polynomial terms to reveal the intricate multivariate interdependence underlying the data’s mean structure. Identifiability issues related to the proposed parameterizations are discussed and expressions to compute maximum likelihood estimates of the parameters of the resulting mean-restricted matrix normal are derived. In order to exploit the proposed parameterizations in a model-based clustering setting, finite mixtures of mean-restricted matrix normals are considered. Integrating structured covariance matrices, the approach maintains model flexibility without succumbing to overfitting. An Expectation-Maximization (EM) algorithm is developed to estimate all model parameters. Through a comprehensive simulation study and a real-world example on climate data, the efficacy of the proposed solutions in capturing complex data relationships is demonstrated, offering significant improvements over traditional methods.</p>