The family of layered thio- and seleno-phosphates has gained attention as potential control dielectrics for the rapidly growing family of two-dimensional and quasi-two-dimensional electronic materials. Here we report a combination of density functional theory calculations, quantum molecular dynamics simulations and variable-temperature, -pressure and -bias piezoresponse force microscopy data to predict and verify the existence of an unusual ferroelectric property-a uniaxial quadruple potential well for Cu displacements-enabled by the van der Waals gap in copper indium thiophosphate (CuInP2S6). The calculated potential energy landscape for Cu displacements is strongly influenced by strain, accounting for the origin of the negative piezoelectric coefficient and rendering CuInP2S6 a rare example of a uniaxial multi-well ferroelectric. Experimental data verify the coexistence of four polarization states and explore the temperature-, pressure- and bias-dependent piezoelectric and ferroelectric properties, which are supported by bias-dependent molecular dynamics simulations. These phenomena offer new opportunities for both fundamental studies and applications in data storage and electronics.