We employ scanning tunneling microscopy (STM) to reveal the structure of DNA-carbon nanotube complexes with unprecedented spatial resolution and compare our experimental results to molecular dynamics simulations. STM images show strands of DNA wrapping around (6,5) nanotubes at approximately 63 degrees angle with a coiling period of 3.3 nm, in agreement with the theoretical predictions. In addition, we observe width modulations along the DNA molecule itself with characteristic lengths of 1.9 and 2.5 nm, which remain unexplained. In our modeling we use a helical coordinate system, which naturally accounts for tube chirality along with an orbital charge density distribution and allows us to simulate this hybrid system with the optimal pi-interaction between DNA bases and the nanotube. Our results provide novel insight into the self-assembling mechanisms of nanotube-DNA hybrids and can be used to guide the development of novel DNA-based nanotube separation and self-assembly methods, as well as drug delivery and cancer therapy techniques.