With the rapid development of the economy and urbanization, the construction of the urban rail transit system has had a great impact on the work, life, and health of residents in buildings along the rail transit line. Thus, it is particularly urgent and necessary to develop base isolation technologies to control and reduce the impact of vibrations of rail transit systems on building structures. High-damping rubber isolation bearings have shown significant effectiveness in the reduction of this impact, and their isolation performance mainly depends on the mechanical and damping energy dissipation characteristics of the high-damping rubber material. This paper aims to investigate the hyper-viscoelastic properties of the high-damping rubber material used for high-damping rubber isolation bearings during the cyclic tension and compression process in the vertical direction. These properties include hyperelastic parameters, viscoelastic coefficients, and the relaxation times of the material. For this purpose, uniaxial cyclic tension and compression tests were conducted. A three-element Maxwell rheological model combining a strain energy density function was proposed for modeling the hyper-viscoelastic behaviors of the materials during the cyclic tension and compression process. Based on the obtained results, an iterative identification procedure was used to determine the constitutive parameters of the material for each loading-unloading cycle. The aforementioned parameters were further expressed as a function of the number of cycles. New insights into hyper-viscoelastic property changes in this high-damping rubber material during the cyclic tension and compression process were gained in this work. These investigations could facilitate the development of computational tools, which would regulate fundamental guidelines for the better controlling and optimization of the isolation performance of the high-damping rubber material used for high-damping rubber isolation bearings, which have a wider perspective of applications in the urban rail transit system.