There are Two ways of doing this (well, 2 simple ways, that is!);
[*] Control the LC with a Single Pole Double Throw - Momentary Contact w/ Center position "off", type toggle switch - most common option,
[*] Use a typical "3-Way Switch" - most simplest.
The type of Contactor you have is a Mechanically Held
(AKA "Latching") Contactor.
One Coil (the "C" Coil), will close the contacts and "Latch" them closed; while the other Coil (the "O" Coil), will "Unlatch" and open the contacts.
The "opposite function" Coil's leads will go through auxiliary contacts of their own - so the coil does not remain continuously energized
(if the Contactor is latched in the "Closed" position, the lead going to the "C" Coil is now opened via the interlocking contact, and the lead going to the "O" Coil will now be closed via interlocked contact - allowing the "O" Coil to be energized when needed).
Simply stated, once the contactor has latched into one position, that coil's circuit will be opened by the interlock contact. At the same time, the opposite function's coil circuit will be closed via the interlock contact.
This makes it possible to leave AC power on the control circuit, without frying the coil - or having a fight breakout between the Open and Close functions!
(Contactor would open and close over and over..... and over....and over.... like the Energizer Rabbit!!!).
Nearly all the Mech. Held LCs I have dealt with have interlocks - and the ASCO brand is no exception.
To verify if your's does have interlocks, energize one of the function coils and leave it hot (keep AC power on) after it latches, or unlatches the contacts.
Feel the coil, to see if it's "Humming".
If no "Hum", coil circuit is "Dead".
FYI: Coil leads ="C"
= "Close" Coil lead terminal,"O"
= "Open" Coil lead terminal,"LINE"
= "Common" terminal for control circuitry
("other" ends of "C" and "O" coils)
For 120 VAC control circuitry, connect the Grounded "Neutral" Conductor to this terminal.