Magnetic field-controlled continuum robots can be steered dexterously within a small dimension. However, its manipulability, generally bending in a C-shape, is not sufficient for most practical applications, e.g., avoiding obstacles to reach the target region, fixing the tip orientation, or passing through sharp access with the lowest damage to the human body. In this paper, a novel magnetic continuum robot (MCR) with multi-mode control using opposite-magnetized magnets is proposed and studied. A prototype robot containing a pair of opposite-magnetized magnets and a hollow elastic tube was designed and constructed. This simple structure does not hinder the miniaturization of robots. The robot is steered by an external mobile magnet, which can generate an inhomogeneous field to offer a varying magnetic torque and force on opposite-magnetized magnets to help achieve multi-mode deformation. A practical mathematical model, coupling point-dipole field model and energy-based kinematics, is developed to describe its multi-mode deformation with a mean error of 0.72 0.52 mm along the arc length, and a mean tip position error of 1.44 0.74 mm. Experimental results show that the opposite-magnetized magnetic continuum robot has a larger tip reachable workspace and improved manipulability compared to other uniform-magnetized MCRs.