The classical Matérn model has been a staple in spatial statistics. Novel data-rich applications in environmental and physical sciences, however, call for new, flexible vector-valued spatial and space-time models. Therefore, the extension of the classical Matérn model has been a problem of active theoretical and methodological interest. In this paper, we offer a new perspective to extending the Matérn covariance model to the vector-valued setting. We adopt a spectral, stochastic integral approach, which allows us to address challenging issues on the validity of the covariance structure and at the same time to obtain new, flexible, and interpretable models. In particular, our multivariate extensions of the Matérn model allow for asymmetric covariance structures. Moreover, the spectral approach provides an essentially complete flexibility in modeling the local structure of the process. We establish closed-form representations of the cross-covariances when available, compare them with existing models, simulate Gaussian instances of these new processes, and demonstrate estimation of the model's parameters through maximum likelihood. An application of the new class of multivariate Matérn models to environmental data indicate their success in capturing inherent covariance-asymmetry phenomena.