This paper proposes a class of reconfigurable deployable polyhedral mechanisms (RDPMs). The reconfigurable property of angulated element is revealed first, and the concept of reconfigurable angulated element (RAE) with four half-platforms is then introduced. The reconfigurable conditions of RAE are derived based on the reconfigurable position (also transition configuration). With RAEs as modules, a method is proposed for constructing RDPMs. Moreover, reconfigurable generalized angulated elements (RGAE) composed of two GAEs are investigated with their reconfigurable conditions identified. Using 1-SE-2-RGAEs, a tetrahedral RDPM with a large magnification ratio is also constructed. Here, SE stands for straight element. The work provides a systematic approach to the design of RDPMs which can switch between two kinds of conventional DMPs.