Gamma and conversion-electron spectra following the I-127(n, gamma) I-128 reaction have been studied with bent-crystal, magnetic, Ge(Li) and Si(Li) spectrometers in the range 20-600 keV. Gamma-gamma prompt and delayed coincidences in the range 25-600 keV have been investigated with the use of two Ge(Li) detectors. The I-128 level scheme involving 40 levels in the energy range 0-1000 keV has been constructed. Parity is determined for all the levels introduced. Unique spin values are assigned to 30 levels. The half-lives of the 133.611, 137.851 and 167.367 keV levels have been measured to be 12 +/- 3, 845 +/- 20 and 175 +/- 15 ns, respectively. The I-127(d, p) I-128 reaction has been investigated with 22 MeV deuterons at an angle of 30-degrees. The I-129(d, t) I-128 reaction has been studied with 15 MeV deuterons at an angle of 30-degrees. The (n,gamma) levels are found to match well with those from the (d,p), (d,t), (n(res), gamma-primary) and (p,n) reactions. A comparison of the (n,gamma) levels with the (d,p) and (d,t) levels enabled the identification of the pi-g7/2nu-s1/2, pi-g7/2nu-d3/2, pi-g7/2nu-h11/2, pi-d5/2nu-s1/2, pi-d5/2nu-d3/2 and pi-d5/2nu-h11/2 two-quasiparticle multiplets. Energy splittings of these pn configurations by residual interactions taken as a combination of short-range Wigner, singlet and tensor forces have been calculated.