The ceramic TbCo0.5Mn0.5O3.07 of double-perovskite structure was prepared by solid-state reaction. Its crystal structure and magnetic and dielectric properties were investigated by first-principles calculations and experimental observations. TbCo0.5Mn0.5O3.07 possesses a monoclinic structure with P2(1)/n space group. The c axis is the easy-magnetization axis, and it is largely caused by Co2+ anisotropy. The predominant valence states are Mn4+ and Co2+, with a small amount of Co3+ coexisting with Co2+. The ordering of Mn4+ and Co2+ results in ferromagnetic Mn4+-Co2+ interactions. Partial disorder of the B-site creates antiferromagnetic Co2+-O-Co2+ or Mn4+-O-Mn4+ interactions. The origin of metamagnetism is associated with the coexistence of antiferromagnetic and ferromagnetic phases. The magnetic exchange bias is strongly dependent on magnetic field, which is considered to be related to the metamagnetic behavior. The possibility of spin glass behavior is excluded by AC susceptibility measurements. The two observed dielectric relaxations are caused by electrons hopping between Co2+ and Mn4+ and between Co3+ and Mn4+.