Chargaff's first parity rule is a property of double-stranded DNA which states that the number of A and T nucleotides, and the number of C and G nucleotides, are the same within the duplex. It arises as a result of the chemistry of nucleic acids which only permits A to bond with T and C to bond with G. In contrast, Chargaff's second parity rule asserts that the same is also true within a single strand, not only for mononucleotide chains, but also for short polynucleotide chains. Unlike the first parity rule, the second is not exact[ 12 ]. Several explanations for the origins of this intrastrand symmetry have been proposed, but the relative contribution of these mechanisms to the symmetry are still not clear[ 19 ]. This work aims to scrutinise Chargaff's second parity rule by developing tests of statistical significance for the rule and then applying them to a large set of bacterial genomes taken from the GenBank repository. We also consider the vector of mononucleotide frequencies (π a : a ∈ {A, C, G, T}), together with the stochastic matrix of conditional nucleotide frequencies P = (P a, b : a, b ∈ {A, C, G, T}), where P a, b is the observed frequency of b-type nucleotides given that they are preceded by an a-type nucleotide, and prove that Chargaff's second parity rule for mononucleotides and dinucleotides is equivalent to P possessing a particular structure. The proposed tests make a uniform assumption either on the set of all 4 × 4 stochastic matrices or on the set of 4-element probability vectors and compare this with the structure induced by Chargaff's second parity rule. When applied to real bacterial genome sequences, the tests generally confirm Chargaff's second parity rule.