The theory of nongravitational interaction between a pressureless dark matter (DM) and dark energy (DE) is a phenomenologically rich cosmological domain which has received magnificent attention in the community. In the present article we have considered some interacting scenarios with some novel features: the interaction functions do not depend on the external parameters of the Universe, rather, they depend on the intrinsic nature of the dark components; the assumption of unidirectional flow of energy between DM and DE has been extended by allowing the possibility of bidirectional energy flow characterized by some sign shifting interaction functions; and the DE equation of state has been considered to be either constant or dynamical in nature. These altogether add new ingredients in this context, and we performed the phase space analysis of each interacting scenario in order to understand their global behavior. According to the existing records in the literature, this combined picture has not been reported elsewhere. From the analyses, we observed that the DE equation of state as well as the coupling parameter(s) of the interaction models can significantly affect the nature of the critical points. It has been found that within these proposed sign shifting interacting scenarios it is possible to obtain stable late time attractors, which may act as global attractors corresponding to an accelerating expansion of the Universe. The overall outcomes of this study clearly highlight that the sign shifting interaction functions are quite appealing in the context of cosmological dynamics, and they deserve further attention.