The Rubik’s Magic puzzle is a mechanical toy comprised of eight thin panels. Connected through a unique cord-based mechanism, the panels can fold along two perpendicular directions, achieving various reconfigurations by switching its rotational axes. This research unravels the hinge-switching process by analysing the puzzle’s stringing patterns and investigating the kinematics of the panel-cord assembly on two thin panels. It can be described by the bifurcation of a spherical 4R linkage. A physical model demonstrates continuous rotation and flipping motion between the pair of thin panels. Furthermore, the study extends its scope by varying panel geometry and thickness on the reconfiguration process. The findings serve as a groundwork for future research in designing cord-based mechanisms to achieve hinge-switching properties for engineering applications.