The level structures of the very neutron-rich nuclei 128Pd and 126Pd have been investigated for the first time. In the r-process waiting-point nucleus 128Pd, a new isomer with a half-life of 5.8(8) μs is proposed to have a spin and parity of 8(+) and is associated with a maximally aligned configuration arising from the g(9/2) proton subshell with seniority υ=2. For 126Pd, two new isomers have been identified with half-lives of 0.33(4) and 0.44(3) μs. The yrast 2(+) energy is much higher in 128Pd than in 126Pd, while the level sequence below the 8(+) isomer in 128Pd is similar to that in the N=82 isotone 130Cd. The electric quadrupole transition that depopulates the 8(+) isomer in 128Pd is more hindered than the corresponding transition in 130Cd, as expected in the seniority scheme for a semimagic, spherical nucleus. These experimental findings indicate that the shell closure at the neutron number N=82 is fairly robust in the neutron-rich Pd isotopes.