We report on the first, to the best of our knowledge, solid-state dual-comb mode-locked laser that simultaneously operates in different dispersion regimes. Due to the intrinsic polarization multiplexing in a birefringent Yb:Ca3NbGa3Si2O14 (Yb:CNGS) gain medium, the laser emits two cross-polarized pulse trains with a repetition rate offset of ∼ 4.8 kHz from a single cavity. We obtain dual pulse generation with a 20-fold difference in duration by setting the net cavity group delay dispersion to cross zero across the emission band of the employed gain medium. While the duration of the soliton-like pulses experiencing anomalous dispersion amounts to 117 fs, the second laser output, which is spectrally located in the normal dispersion region, is strongly chirped with a pulse duration of 2360 fs.