Tension-compression operation in MR elastomers (MREs) offers both the most compact design and superior stiffness in many vertical load-bearing applications such as MRE bearing isolators in bridges and buildings, suspension systems and engine mounts in cars, and vibration control equipment. It suffers, however, from lack of good computational models to predict device performance, and as a result shear-mode MREs are widely used in the industry, despite their low stiffness and load-bearing capacity. We start with a comprehensive review of MREs modeling and their dynamic characteristics, showing previous studies have mostly focused on dynamic behavior of MRE in the shear mode, though the MRE strength and MR effect are greatly decreased at high strain amplitudes, due to increasing distance between the magnetic particles. Moreover, the characteristic parameters of the current models either frequency, strain or magnetic field are constant; hence, new model parameters must be recalculated for new loading conditions. This is an experimentally time consuming and computationally expensive task, and no models capture the full dynamic behavior of the MREs at all loading conditions. In this study, we present an experimental setup to test MREs in a coupled tension-compression mode, as well as a novel phenomenological model which fully predicts the stress-strain behavior of the as a function of magnetic flux density, loading frequency and strain. We use a training set of experiments to find the experimentally derived model parameters, which can predict by interpolation the MRE behavior in the relatively large continuous range of frequency, strain and magnetic field. We also challenge the model to make extrapolating predictions and compare to additional experiments outside the training experimental data set with good agreement. Further development of this model would allow design and control of engineering structures equipped with tension-compression MREs and all the advantages they offer.