Van der Waals "sliding" ferroelectric bilayers, whose electric polarization is locked to the interlayer alignment, show promise for future non-volatile memory and other nanoelectronic devices. These applications require a fuller understanding of the polarization stability and switching properties, which present models have described in terms of an Ising-like binary polarization. However, it is a much larger translation symmetry that is broken in the polar state. Here we introduce a discrete statistical-mechanical model that emphasizes the effect of this larger symmetry. Through Monte-Carlo numerics we show this model possesses a richer phase diagram, including an intermediate critical phase of algebraically-correlated polarization. A low energy effective theory allows us to connect the ferroelectric-paraelectric transition to the Berezinskii-Kosterlitz-Thouless class, driven by excitations not available in Ising-like models. Our results indicate the need for theoretical models of this ferroelectric system to account for the larger symmetry.