We consider theoretically materials whose electromagnetic properties possess parity-time (PT) symmetry and are periodic in two dimensions. When designed for optical frequencies such structures are commonly known as two-dimensional (2D) photonic crystals. With the addition of PT symmetry the optical modes of 2D photonic crystals exhibit thresholdless spontaneous PT-symmetry breaking near the Brillouin zone boundary, which is analogous to what has previously been studied in PT-symmetric structures with one-dimensional periodicity. Consistent with previous work, we find that spontaneous PT-symmetry breaking occurs at band crossings in the photonic dispersion diagram. Due to the extra spatial degree of freedom in 2D periodic systems, their band structures contain more band crossings and higher-order degeneracies than their one-dimensional counterparts. This work provides a comprehensive theoretical analysis of spontaneous PT-symmetry breaking at these points in the band structure. We find that, as in the case of one-dimensional structures, photonic band gaps exist at k=0. We also find that at points of degeneracy with order higher than 2, bands merge pairwise to form broken-PT-symmetry supermodes. If the degeneracy order is even, this means multiple pairs of bands can form distinct (nondegenerate) broken-symmetry supermodes. If the order of degeneracy is odd, at least one of the bands will have protected PT symmetry. At other points of degeneracy, we find that the PT symmetry of the modes may be protected and we provide a spatial mode symmetry argument to explain this behavior. Finally, we identify a point at which two broken-PT-symmetry supermodes become degenerate, creating a point of fourfold degeneracy in the broken-PT-symmetry regime.