In this article, we introduce a cycloidal legs-augmented wheel (CLAW), a novel wheel-leg mechanism for stair and obstacle climbing. CLAW integrates three leg segments with a wheel using a specialized passive bar mechanism inspired by a cycloidal rotor design. These legs extend outward to reach heights from the front and retract back within the wheel boundary as they approach the ground. This unique mechanism enables the legs to generate cycloidal trajectories as the wheel rotates without additional actuators, thereby preserving the operation and control simplicity of conventional wheeled robots while significantly improving climbing abilities. Experimental tests with CLAWbot, a fixed-axis four-wheeled robot equipped with the CLAW mechanisms, demonstrated obstacle climbing up to 2.6 times the wheel radius and reliable traversing of different staircases, obstacles, and diverse terrains, such as a concrete floor, grass, a ramp, and rocky surfaces. It also exhibited smooth trajectories during turns and climbing.