Abstract We show that the potential of Nambu-Goldstone bosons can have two or more local minima e.g. at antipodal positions in the vacuum manifold. This happens in many models of composite Higgs and of composite Dark Matter. Trigonometric potentials lead to unusual features, such as symmetry non-restoration at high temperature. In some models, such as the minimal SO(5)/SO(4) composite Higgs with fermions in the fundamental representation, the two minima are degenerate giving cosmological domain-wall problems. Otherwise, an unusual cosmology arises, that can lead to supermassive primordial black holes; to vacuum or thermal decays; to a high-temperature phase of broken SU(2) L , possibly interesting for baryogenesis.