A valve-operating mechanism for an internal combustion engine having intake and exhaust valves and valve control members for controlling the intake and exhaust valves. The intake and exhaust valves are configured as dual valves having an outer valve and an inner valve. The valve control members, at least during scavenging, simultaneously close the outer valves of the intake and exhaust valves, producing a non-overlapping state, and simultaneously open the inner valves of the intake and exhaust valves, producing an overlapping state.