A centroidal Voronoi tessellation (CVT) is a special kind of Voronoi tessellation such that the generating points of the tessellation are also the mass centroids of the corresponding regions. Due to their innate optimization properties, CVTs have applications in diverse fields; however, the theoretical nature of these tessellations is far from well understood. We approach some open questions about CVTs by looking in particular at 2-point tessellations of regular polygons with constant density. We show that for any CVT of an even-sided polygon, the Voronoi boundary or the centroids must lie on lines of symmetry, and anticipate that the same is true for odd-sided polygons. We predict which of these configurations are stable under Lloyd's algorithm for computing CVTs. Finally, we illustrate basins of attraction for the stable CVTs under Lloyd's algorithm.