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By applying a quasi-static transverse load, shallow cylindrical shells can be snapped between two inverted stable states. This dynamic transition, initiated at either a limit point (fold) or a subcritical bifurcation, traverses a region of instability. The ensuing large displacements are attractive for shape adaptation of multi-functional engineering structures. Previous research on isotropic cylindrical shells has indicated the presence of isolated regions of stability in the otherwise unstable transition region. While these regions are theoretically attractive for multi-stage snap-morphing, they are difficult to attain in practice by means of a single control parameter. In this paper, we study the effect of orthotropic material properties on the structural stability and snap-through behavior of shallow cylindrical shells. In addition, we explore complex stability phenomena created by material orthotropy, which are attractive for multi-stage morphing. The problem is analyzed in a robust manner using a technique known as generalized path-following, which combines the mathematical domains of finite element analysis and numerical continuation. The present study shows, in particular, that laminates comprising layers of different directional material properties provide the ability to tailor the elastic bifurcation behavior of cylindrical shells. For example, the lamination scheme can be varied to add isolated regions of stability or entirely remove them; to induce snaking that allows transitioning between the two inverted cylindrical shapes through a series of snaps; and finally, to create groupings of multiple unique stable configurations that can be attained via additional shape control.