The neutral safety switch provides a ground for the starter relay when the transmission is in neutral or park but opens the ground circuit in reverse, drive, 2nd gear and low gear so that the starter motor can't be engaged.
In your particular case, there is no park function so the engine will only be able to be started in neutral and a good parking brake system (or a set of chock blocks) is a must for this vehicle.
....and you should be able to tow the car to get it started. You just have to be going fast enough. The old cast iron transmissions had rear pumps and could be used to spin the engine over if you towed the vehicle fast enough.
I seem to remember going about 25 mph behind my dad's '60 Windsor and dumping the trans in my '55 DeSoto into drive and the engine finally sputtered to life once the fuel got up to the carb from the tank. Remembering the actual speed is a bit tough because we did that in 1963 after buying the car from my dad's friend's used car lot so that I could pull the engine and dump it in my '39 Dodge pickup. The 6 volt battery was on it's last legs and we couldn't get the engine to fire so my dad just pulled out the tow rope and off we went.
There is a quadrant inside the transmission with detents on it that operates the neutral safety switch.
If someone has been fiddling with the valve body and installed it back in the transmission without, first, removing the neutral safety switch one of two things can happen. The quadrant will get bent, or in worst case, break off and either scenario will render the neutral safety switch inoperative.
The early sixties were transition years for ChRyCo starter motor circuitry. Chryslers, Imperials and DeSotos used a separate relay along with a starter-mounted solenoid.
Dodges and Plymouths used a solenoid similar to the units commonly found of FoMoCo products with one important difference and that is in the functionality of the two smaller pins on the solenoid.
If it's got an "I" and an"S" cast into the housing, it's a Ford starter solenoid, not a Chrysler solenoid. and will, most likely, not work because the I and S terminals aren't on each end of the "pick" coil.
This solenoid coil is NOT internally grounded and needs both a power and a ground connection in order to operate. One terminal gets connected to the ignition switch "start" wire and the other terminal is connected to the neutral safety switch which will provide a ground in, either, the neutral or park positions.
This solenoid is NOT the same as identically-shaped similar FoMoCo solenoids that DO have the "pick" coil internally grounded and are energized by simply applying power to the S terminal.
The I terminal is used as a ballast resistor bypass terminal (to provide a full 12 volts to the coil when starting the engine) that is only live when the solenoid was activated.
GM utilizes the same technology in their starter circuitry wherein the R terminal on the starter solenoid is connected to the distributor coil side of the ballast resistor to boost the voltage to the ignition coil during the engine starting process.
Make sure you have the correct type of solenoid. A FoMoCo style solenoid won't provide any neutral safety function. It WILL engage the starter regardless of the state of the transmission as it is an internally-grounded solenoid and as soon as you apply power to the "S" terminal, the solenoid will engage the starter.
If you can make the solenoid operate by applying 12 volts to one small terminal and ground to the other terminal WITH THE SOLENOID ON A BENCH AND NOT, OTHERWISE, GROUNDED, the solenoid WILL function as a neutral safety device.
If the only way you can get the solenoid to operate is to apply 12 volts to the S terminal and GROUND THE MOUNTING BRACKET, then you will NOT have neutral safety protection with that solenoid.
On the late fifties and early sixties (and beyond) Chryslers, Imperials and DeSotos, the neutral safety switch is connected to a starter RELAY which, in turn, is connected to the starter solenoid (which is mounted on the starter itself). Later on, this relay was incorporated in all ChRyCo cars.
The "SOL" terminal connects to the starter solenoid
The "I" terminal is connected to the start post on the ignition switch
The bullet terminal is connected to the neutral safety switch on the transmission.
"BAT" is connected to the battery and also to the starter solenoid.
When power from the ignition switch is applied to the "I" terminal AND the neutral safety switch is also providing a ground to the bullet terminal, the relay will engage and provide 12 volts on the "SOL" wire to activate the starter-mounted solenoid.
