Despite our long familiarity with how the genetic code specifies the amino acid sequence, we still know little about why it is organized in the way that it is. Contrary to the view that the organization of the genetic code is a "frozen accident" of evolution, recent studies have demonstrated that it is highly nonrandom, with implications for both codon assignment and usage. We hypothesize that this inherent nonrandomness may facilitate the coexistence of both sequence and structural information in DNA. Here, we take advantage of a simple metric of intrinsic DNA flexibility to analyze mutational effects on the four phosphate linkages present in any given codon. Application of a simple evolutionary neutral model of substitution to random sequences, translated with alternative genetic codes, reveals that the standard code is highly optimized to favor synonymous substitutions that maximize DNA polymer flexibility, potentially counteracting neutral evolutionary drift toward stiffer DNA caused by spontaneous deamination. Comparison to existing mutational patterns in yeast also demonstrates evidence of strong selective constraint on DNA flexibility, especially at so-called "silent" sites. We also report a fundamental relationship between DNA flexibility, codon usage bias, and several important evolutionary descriptors of comparative genomics (e.g., base composition, transition/transversion ratio, and nonsynonymous vs. synonymous substitution rate). Recent advances in structural genomics have emphasized the role of the DNA polymer's flexibility in both gene function and whole genome folding, thereby implicating possible reasons for codons to facilitate the multiplexing of both genetic and structural information within the same molecular context.