The ground state and magnetization process of an exactly solved spin-$1/2$ Ising-Heisenberg orthogonal-dimer chain with two different gyromagnetic factors of the Ising and Heisenberg spins are investigated in detail. It is shown that the investigated quantum spin chain exhibits up to seven possible ground states depending on a mutual interplay of the magnetic field, intra- and inter-dimer coupling constants. More specifically, the frustrated and modulated quantum antiferromagnetic phases are responsible in zero-temperature magnetization curves for a zero magnetization plateau. The intermediate 1/11- and 5/11-plateaus emerge due to the frustrated and modulated quantum ferrimagnetic phases, while the intermediate 9/11- and 10/11-plateaus can be attributed to the quantum and classical ferrimagnetic phases. It is conjectured that the magnetization plateau experimentally observed in a high-field magnetization curve of 3$d$-4$f$ heterobimetallic coordination polymer [{Dy(hfac)$_2$(CH$_3$OH)}$_2${Cu(dmg)(Hdmg)}$_2$]$_n$ (H$_2$dmg $=$ dimethylglyoxime; Hhfac $=$ 1,1,1,5,5,5-hexafluoropentane-2,4-dione) could be attributed to the classical and quantum ferrimagnetic phases.
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