In the odd-mass 113-119Sb and 117-127I nuclei, bands with ΔJ = 1 spacing have been observed built on a low-lying 9/2+ state. We have described these bands as resulting from a band-mixing calculation at the equilibrium deformation corresponding with the 9/2+ Nilsson orbital. Total potential energy surfaces, energy spectra as well as electromagnetic properties for these nuclei are presented and compared with the experimental data. Failure in describing the observed phenomena within a harmonic core coupling model is pointed out. Therefore hole-anharmonic core coupling calculations have been performed in order to obtain the ΔJ = 1 band structure as well as improved intensity rules. Results from both the band-mixing and hole-anharmonic core coupling calculations are compared and similarities are pointed out.