This paper presents the first combined measurement of the double-differential muon neutrino and antineutrino charged-current cross sections with no pions in the final state on hydrocarbon at the off-axis near detector of the T2K experiment. The data analyzed in this work comprise 5.8$\times$10$^{20}$ and 6.3$\times$10$^{20}$ protons on target in neutrino and antineutrino mode respectively, at a beam energy peak of 0.6 GeV. Using the two measured cross sections, the sum, difference and asymmetry were calculated with the aim of better understanding the nuclear effects involved in such interactions. The extracted measurements have been compared with the prediction from different Monte Carlo generators and theoretical models showing that the difference between the two cross sections have interesting sensitivity to nuclear effects.