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In this work, we propose a class of ferroelectrics (which we denote "double-path" ferroelectrics), characterized by two competing polarization switching paths for which the change in polarization is different and in fact of opposite sign. Depending on which path is favorable under given conditions, this leads to different identification of up- and down-polarized states. Since the sign of piezoelectric response depends on the assignment of up- or down-polarized state for a specific structure, this means that the material can exhibit different signs of the piezoelectric response under different conditions. We focus on HfO$_2$ as a key example. Our first-principles calculations show that there are two competing paths in HfO$_2$, resulting from different displacements of the atoms from the initial to the final structures, and the change in polarization along these two paths is of opposite sign. These results provide a natural explanation for the recently observed discrepancy in the signs of piezoelectric responses in HfO$_2$ between theoretical first-principles calculations and experimental observation. Further, this allows predictions of how to favor one path over another by changes in conditions and compositional tuning. This family of materials also includes other candidates, such as CuInP$_2$S$_6$ and theoretically proposed LaVO$_3$-SrVO$_3$ superlattice. We finally note that double-path ferroelectrics possess novel electromechanical properties since the signs of their piezoelectric responses can be switched.