Does the engine and electric motor have to sync in some way? My 2006 Prius will jump forward when the engine starts when I'm sitting at a stop sign or red light.
The gears between them are never disengaged. They can't get out of 'sync'. Sitting in the garage or driveway, mine also feels like it is trying to move slightly when the ICE fires up, even in Park. This is normal Toyota HSD behavior. It doesn't feel totally seamless, but you can ignore it and nothing bad happens.
The car's computers decide the most efficient amount of electric motor power and gas engine power to use in each specific driving scenario.
Since the engine is started by one of the motors in the transmission and nothing in the transmission is ever disengaged, starting the engine does rock the car a little. It's especially noticeable if you're sitting stopped without holding the brake, so the car can rock more easily. They've always tried to send the right current to MG2 in the other direction to cancel out what MG1 is doing to crank the engine, but the canceling is only kind of approximate. My Gen 3, I notice, will even apply the brakes by itself, a split second before starting the engine, if I am not holding the brakes myself. Clever new trick; my Gen 1 didn't do that. Whether it started with Gen 2 or not until Gen 3, I'm not sure.