Let C(X) be the hyperspace of all subcontinua of a (metric) continuum X. It is known that C(X) is homogeneous if and only if C(X) is the Hilbert cube. We are interested in knowing when C(X) is 1/2-homogeneous, meaning that there are exactly two orbits for the action of the group of homeomorphisms of C(X) onto C(X). It is shown that if X is a locally connected continuum or a nondegenerate atriodic continuum, and if C(X) is 1/2-homogeneous, then X is an arc or a simple closed curve. We do not know whether an arc and a simple closed curve are the only continua X for which C(X) is 1/2-homogeneous.