No, the motor slows as the transmitter lever approaches the centre position and then stops when it gets there. If the lever movement is continued in the same direction the motor movement is reversed. The creeping problem caused by the very narrow deadband, mentioned by Malcolm, is a real pain. I modified two servos like this but have decided to by-pass their electronics and use FLJ's P44 units. In other words, I'm only going to use the servo's as sources of geared motors. The disadvantage is that there will be no speed control. My application is operating a winch on a boat.
To explain it from a slightly different view - the electronics compares the position pot to the required position given from the control stick, usually via the radio. If there is a difference, it drives the motor to correct this difference, the bigger the difference, the harder the drive, and vice-versa, so that the motor drive is reduced the nearer correct it is. If the motor drives the pot, a new position is taken up. If not, there is a de facto speed controller.
In the elder days, the same chip was used for both functions,the difference being the component values that set the deadband. The datasheets for the ZN409 were fairly comprehensive, but could cause sore brains. They did it for me, anyway.
Just been warned about Frankies post - the best bet there would probably be to gear up the output of a standard, out of the box, servo from 90 to 360 or more degrees. This would have the disadvantage of being a bit (!) fiddly around reverse, but would leave the setup capable of returning to dead ahead while having the lever give some indication as to which way the drive was pointing.