Gamma-ray decays from states in ¹²⁹Ce have been studied using ¹⁶O-induced reactions on ¹¹⁶Sn and ¹¹⁷Sn targets. Previously known positive- and negative-parity bands have been extended to spins of ³¹/₂ and ³⁹/₂, respectively. Band crossings are seen for the first time in ¹²⁹Ce and occur near the same rotational frequency as the h₁₁2/ proton band crossing in ¹³⁰Ce. The signature splittings in the bands are discussed in terms of the cranked shell model with a prolate deformation epsilon ₂=0.25. The data show that the one-quasiparticle bands based on the h_11/2 (523) 7/2 Nilsson orbit have gamma approximately -15 degrees (signature alpha =+¹/₂) and gamma approximately -30 degrees ( alpha =-¹/₂). After the band crossing, the three-quasiparticle bands (including two h_11/2 quasiprotons) have gamma approximately 0 degrees for both signatures. The positive-parity states corresponds to gamma approximately 0 degrees for both the one-quasiparticle and three-quasiparticle bands. The branching ratios have been used to deduce the ratio (g_K-g_R)/Q₀ as a function of rotational frequency. This ratio shows an increase when the backbending takes place.