A novel twist morphing concept is explored, that exploits the buckling instabilities of slender spar webs integrated into a wing structure to control the twisting response to external loads. The main novelty of the concept is in controlling the effective shear stiffness of the post-buckled slender spar webs through external variable constraints acting on the out-of-plane buckling deformations. A methodology for the design and analysis of these novel morphing structures is proposed and implemented in the design of a wing box structure of promising twist morphing capabilities. The overall design process is structured into a multilevel process of increased complexity. In the first level, the morphing structure is simplified to a wing box with slender spar webs. With the objective of maximizing the morphing twists that can be achieved under the action of an external quasi-static torque, the wing box design space is explored in terms of its cross-sectional dimensions and the material assigned to the slender spar webs. In the second level, the morphing structure is expanded to include both the wing box and the external devices required to implement the adaptive constraints acting on the slender spar webs' out-of-plane buckling deformations. At this level, the objective is to design adaptive constraining devices that maximize the twist morphing capabilities, for which the influence of the constraining devices over the twisting response and their effectiveness in restraining the slender spar webs' buckling deformations become the main concerns. After an extensive design process, a design solution for the adaptive constraining devices is proposed, for which thorough analyses on the twist morphing capabilities are performed. In addition, the sensitivity of the twist morphing capabilities to the slender spar webs' geometrical imperfections is investigated.