Rigid meromorphic cocycles are defined in the setting of orthogonal groups of arbitrary real signature and constructed in some instances via a p-adic analogue of Borcherds' singular theta lift. The values of rigid meromorphic cocycles at special points of an associated p-adic symmetric space are then conjectured to belong to class fields of suitable global reflex fields, suggesting an eventual framework for explicit class field theory beyond the setting of CM fields explored in the treatise of Shimura and Taniyama.