Identify a relay?

[kwired]

The existing relays control 3 light circuits with 2 sets three-way switches. The lighting is being replaced (including the circuits). It would be convenient to re-use the control.

Verify what you have there. What looks like a three way switch is probably SPDT - but with a spring return to a neutral position and a circuit similar to what Gar described.

 

[olc]

Sorry I was assuming electric maintained relay. They could be momentary contact. I wasn't paying that much attention to the switches other than there are in 3 locations in the (large) room.
There will be dimmers but they will be on the 120V light circuits (load side of any control).