The presence of circular codes in mRNA coding sequences is postulated to be involved in informational mechanisms aimed at detecting and maintaining the normal reading frame during protein synthesis. Most of the recent research is focused on trinucleotide circular codes. However, also dinucleotide circular codes are important since dinucleotides are ubiquitous in genomes and associated to important biological functions. In this work we adopt the group theoretic approach used for trinucleotide codes in Fimmel et al. (2015) to study dinucleotide circular codes and highlight their symmetry properties. Moreover, we characterize such codes in terms of n-circularity and provide a graph representation that allows to visualize them geometrically. The results establish a theoretical framework for the study of the biological implications of dinucleotide circular codes in genomic sequences.