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In order to explore the properties of a two-sublattice ferroelectric, we measured the infrared and Raman scattering response of CuInP2S6 across the ferroelectric and glassy transitions and compared our findings to a symmetry analysis, calculations of phase stability, and lattice dynamics. In addition to uncovering displacive character and a large hysteresis region surrounding the ferroelectric transition temperature T_C, we identify the vibrational modes that stabilize the polar phase and confirm the presence of two ferroelectric variants with opposite polarizations. Below TC, a poorly understood relaxational or glassy transition at Tg is characterized by local structure changes in the form of subtle peak shifting and activation of low frequency out-of-plane Cu- and In-containing modes. The latter are due to changes in the Cu/In coordination environments and associated order-disorder processes. Moreover, Tg takes place in two steps with another large hysteresis region and significant underlying scattering. Combined with imaging of the room temperature phase separation, this effort lays the groundwork for studying CuInP2S6 under external stimuli and in the ultra-thin limit.