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Although robots are increasingly expected to perform inspection tasks in three-dimensional ferromagnetic structural environments, magnetic-wheeled climbing robots face significant challenges in overcoming obstacles and transiting between planes. In this paper, we propose a novel bicycle-like magnetic-wheeled climbing robot, named BiMagBot, featuring two magnetic wheels that allow the adaptive adjustment of magnetic adhesion without the need for active control. The front wheel incorporates an arc tentacle mechanism that rotates a ring magnet to adjust the magnetic adhesion, while the rear wheel uses an eccentric shaft-hole design to facilitate a smooth transition of magnetic adhesion between surfaces. The magnetic forces acting on both wheels during transitions through concave corners were analyzed and discussed via simulations to elucidate the underlying principles. A prototype of the robot was developed and tested experimentally. The results show that the front and rear wheels can adjust the magnetic adhesion during the transition of corners with angles ranging from 90° to 315°. The robot only weighs 1.6 kg, but it can carry a weight of 2 kg with a speed of 0.9 m/s to transit across concave corners, demonstrating comprehensive capabilities in plane transition, ease of control, and load capacity.