A spark-ignition direct injection engine is provided. The engine includes an engine body having a cylinder, intake and exhaust valves for opening and closing intake and exhaust ports that open to the cylinder, respectively, a fuel injection valve for injecting fuel into the cylinder, a catalyst in an exhaust passage, and a controller for operating the engine body by controlling the intake and exhaust valves and the fuel injection valve. The controller performs a fuel cut by suspending the fuel injection during engine deceleration in a non-operated accelerator state, and when the engine speed drops to a predetermined speed, the controller resumes the fuel injection. During the fuel cut, the controller opens the exhaust valve on an intake stroke and then opens the intake valve, and after closing the exhaust valve and when mixture gas is blown back to the intake port, the controller closes the intake valve.