An assembly is provided for a gas turbine engine with an axial centerline. This assembly includes a gearbox, a first torque transmission apparatus and a second torque transmission apparatus. The gearbox includes a plurality of first gears and a plurality of second gears. The first gears are meshed together and respectively rotatable about parallel first gear axes. The second gears are meshed together and respectively rotatable about parallel second gear axes. Each of the first gear axes and each of the second gear axes is non-parallel with the axial centerline. The first torque transmission apparatus is configured to drive the first gears. The second torque transmission apparatus is configured to drive the second gears.