The level structures of the very neutron-rich nuclei 128Pd82 and 126Pd80 have been investigated for the first time. A new isomer with a half-life of 5.8(8) μs in 128Pd is proposed to have a spin and parity of 8+ and is associated with a maximally aligned configuration arising from the g9/2Pd proton subshell with seniority υ = 2. The level sequence below the 8+ isomer is similar to that in the N = 82 isotone 130Cd, but the electric quadrupole transition that depopulates the 8+ isomer is more hindered in 128Pd than in 130Cd, as expected in the seniority scheme for a semi-magic, spherical nucleus. For 126Pd, three new isomers with Jπ = (5-), (7-), and (10+) have been identified with half-lives of 0.33(4) μs, 0.44(3) μs, and 23.0(8) ms, respectively. The smaller energy difference between the 10+ and 7- isomers in 126Pd than in the heavier N = 80 isotones can be interpreted as being ascribed to the monopole shift of the h11/2 neutron orbit. The nature of the N = 82 shell closure scrutinized with these characteristic isomers is discussed.