The high-spin states of $^{123,125,127}\mathrm{I}$ have been investigated via the $^{A}\mathrm{Sn}(^{6}\mathrm{Li}, 3n)^{A+3}\mathrm{I}$ reactions to study the collective properties of the odd-mass I isotopes. In-beam measurements of $\ensuremath{\gamma}$-ray excitations, $\ensuremath{\gamma}\ensuremath{-}\ensuremath{\gamma}$ coincidences, $\ensuremath{\gamma}$-ray angular distributions, and pulsed beam-$\ensuremath{\gamma}$ timing were performed with Ge detectors to determine level energies, decay schemes, $\ensuremath{\gamma}$-ray multipolarities, ${J}^{\ensuremath{\pi}}$ assignments, and lifetime information. A similar study of the $^{117,119,121}\mathrm{I}$ isotopes is reported in the following paper. Two collective features have been identified in these odd-mass I nuclei. Systematic $\ensuremath{\Delta}J=1$ bands built on low-lying ${\frac{9}{2}}^{+}$ proton-hole (4p-1h) states were observed. The ${\frac{9}{2}}^{+}$ bandheads, that involve the excitation of a $1{g}_{\frac{9}{2}}$ proton across the $Z=50$ shell, drop to very low energies near the middle of the neutron shell. The properties of the ${\frac{9}{2}}^{+}$ proton-hole states for all of the odd-mass I isotopes are presented and related to the systematic information for the proton-hole states in the entire $Zg50$ transition region. Systematic $\ensuremath{\Delta}J=2$ bands built on ${\frac{11}{2}}^{\ensuremath{-}}$ ($1{h}_{\frac{11}{2}}$ quasiproton) states, on ${\frac{7}{2}}^{+}$ ($1{g}_{\frac{7}{2}}$ quasiproton) states, and on ${\frac{5}{2}}^{+}$ ($2{d}_{\frac{5}{2}}$ quasiproton) states were also observed. The $\ensuremath{\Delta}J=2$ band spacings generally follow the spacings of the Te-core ground-state bands with the exception of the ${\frac{11}{2}}^{\ensuremath{-}}$ $\ensuremath{\Delta}J=2$ bands, for which the spacings decrease significantly relative to those for the Te cores as $A$ decreases. These systematic properties are discussed in terms of several theoretical approaches to the onset of collectivity in transitional nuclei. An isomer at 2660 keV in $^{123}\mathrm{I}$ was observed to have a mean lifetime $\ensuremath{\tau}=38\ifmmode\pm\else\textpm\fi{}3$ ns.NUCLEAR REACTIONS $^{120\ensuremath{-}124}\mathrm{Sn}(^{6}\mathrm{Li}, 3n)^{123\ensuremath{-}127}\mathrm{I}$; measured $\ensuremath{\gamma}\ensuremath{-}\ensuremath{\gamma}$ coincidences, $\ensuremath{\gamma}(E, \ensuremath{\theta}, t)$; deduced level schemes in odd-mass $^{123\ensuremath{-}127}\mathrm{I}$, $\ensuremath{\gamma}$ multipolarities, ${J}^{\ensuremath{\pi}}$, ${T}_{\frac{1}{2}}$. Enriched targets, Ge(Li) detectors.