We study the orientation dynamics of two-dimensional concavo-convex solid bodies that are denser than the fluid through which they fall under gravity. We show that the orientation dynamics of the body, quantified in terms of the angle π relative to the horizontal, undergoes a transcritical bifurcation at a Reynolds number and a subcritical pitchfork bifurcation at a Reynolds number . For , the concave-downwards orientation of π=0 is unstable and bodies overturn into the π=π orientation. For , the falling body has two stable equilibria at π=0andπ=π for steady descent. For , the concave-downwards orientation of π=0 is again unstable and bodies that start concave-downwards exhibit overstable oscillations about the unstable fixed point, eventually tumbling into the stable π=π orientation. The at which the subcritical pitchfork bifurcation occurs is distinct from the π β’π for the onset of vortex shedding, which causes the π=π equilibrium to also become unstable, with bodies fluttering about π=π. The complex orientation dynamics of irregularly shaped bodies evidenced here are relevant in a wide range of settings, from the tumbling of hydrometeors to the settling of mollusk shells.