This paper presents a comprehensive constitutive framework for the macroscopic, continuum-based modeling of isotropic, compressible magneto-active polymers (MAPs) in the magnetic linear regime at finite strains. An energy-based computational framework is applied to compute the homogenized macro-response of the magneto-active composite. This methodology allows the in silico generation of a comprehensive data set and overcomes difficulties in the material characterization of MAPs based on experiments due to strong macrostructural shape effects. We outline basic ingredients of a suitable constitutive macroscale model in an energy-based setting and discuss details of an adequate fitting algorithm for the accurate parameter identification of the developed macroscale model based on the generated homogenization data set. The performance of the developed macroscale model is demonstrated by solving some application-oriented boundary value problems. The main emphasis of the numerical studies lies on the multiscale analysis with a micro–macro decoupling scheme as well as the investigation of macrostructural shape effects.