The single-stranded DNA to single-walled carbon nanotube (SWCNT) hybrid continues to attract significant interest as an exemplary biological molecule-nanomaterial conjugate. In addition to their many biomedical uses, such as in vivo sensing and delivery of molecular cargo, DNA-SWCNT hybrids enable the sorting of SWCNTs according to their chirality. Current experimental methods have fallen short of identifying the actual structural ensemble of DNA adsorbed onto SWCNTs that enables and controls several of these phenomena. Molecular dynamics (MD) simulation has been a useful tool for studying the structure of these hybrid molecules. In recent studies, using replica exchange MD (REMD) simulation we have shown that novel secondary structures emerge and that these structures are DNA-sequence and SWCNT-type dependent. Here, we use REMD to investigate in detail the structural characteristics of two DNA-SWCNT recognition pairs: (TAT)(4)-(6,5)-SWCNT, i.e., DNA sequence TATTATTATTAT bound to the (6,5) chirality SWCNT, and (CCG)(2)CC-(8,7)-SWCNT as well as off-recognition pairs (TAT)(4)-(8,7)-SWCNT and (CCG)(2)CC-(6,5)-SWCNT. From a structural clustering analysis, dominant equilibrium structures are identified and show a right-handed self-stitched motif for (TAT)(4)-(6,5) in contrast to a left-handed beta-barrel for (CCG)(2)CC-(8,7). Additionally, characteristics such as DNA end-to-end distance, solvent accessible SWCNT surface area, DNA hydrogen bonding between bases, and DNA dihedral distributions have been probed in detail as a function of the number of DNA strands adsorbed onto the nanotube. We find that the DNA structures adsorbed onto a nanotube are also stabilized by significant numbers of non-Watson-Crick hydrogen bonds (intrastrand and interstrand) in addition to pi-pi stacking between DNA bases and nanotube surface and Watson-Crick pairs. Finally, we provide a summary of DNA structures observed for various DNA-SWCNT hybrids as a preliminary set of motifs that may be involved in the functional role of these hybrids.