Three three-quasiparticle isomers, one at an excitation energy of 2.3 MeV with T1/2=0.48(4)μs, and two shorter-lived with unknown half-lives at slightly lower energies have been identified in Nd129 using the MARA + JUROGAM 3 setup and the recoil tagging technique. All three isomers present decay patterns characteristic of high-K isomers. The known 6.7 s β-decaying isomer previously assigned to the 5/2+ level is now assigned to the new 7/2- ground state. A new low-spin 5/2+ isomeric state with a half-life of a few tens of nanoseconds has been identified, while a previously known 2.6 s β-decay activity was assigned to the band head of the ν1/2+411 band. The transitions depopulating the high-K isomers to low-lying states also establish the relative energies of three low-lying one-quasiparticle bands, leading to a new spin-parity assignment of 7/2- to the ground state of Nd129. The partial half-lives of the depopulating transitions suggest spin-parities 21/2+, 19/2+, and 17/2+ for the three high-K isomers. The properties of the band built on the 21/2+ isomeric state suggest a one neutron-two proton configuration. Based on the results of extensive calculations with different models, we also assign one neutron-two proton configurations to the 19/2+ and 17/2+ isomeric states. The assigned configurations of the 17/2+ and 21/2+ isomeric states involve the π9/2+404 orbital, which is identified in three-quasiparticle bands of proton-rich A≈130 nuclei.