The DNA-single-walled carbon nanotube (SWCNT) hybrid molecule has attracted significant attention recently for its ability to disperse and sort SWCNTs according to their chirality. Key for utilizing their unique properties is an understanding of the structure of DNA adsorbed on the SWCNT surface which we study here using molecular simulations. Using replica exchange molecular dynamics (REMD), we explore equilibrium structures formed by single strands of 12-mer oligonucleotides, of varying sequence, adsorbed on a (6,5)-SWCNT. We find a consistent motif in which the DNA strand forms a right-handed helical wrap around the SWCNT, stabilized by "stitches" (hydrogen bonding between distant bases) to itself. Variability among equilibrium populations of DNA self-stitched structures was observed and shown to be directly influenced by DNA sequence and composition. Competition between conformational entropy and hydrogen bonding between bases is predicted to be responsible for the formation of random versus stitched configurations.