How do they maintain state without using power?

(Mechanically) latching relays use a solenoid operated bistable mechanism. There are variety of schemes to do this bit the all offer the functionality of switching between to states when activated—think of a retractable pen that moves in and out alternately with the press of one button.

 

DigiKey has ~360 different latching Relays if you'd like to use just 1 Relay.
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just to add on LowQCab's suggestion... all those latching relays, inside are usually a DPDT relay with a simple wiring. You can make the exact same schematics if you dont have the latching relay availible in your country. For example, for me it is impossible to order from Digikey but it is really simple to make a self-latching relay with a standard DPDT:


Also check my post #80 on the previous page.

 

If a push button with a normally closed contact (SPDT) is available then a much simpler scheme becomes available, except the operation depends on the length of time the button is held depressed. A longer time to switch off, a shorter time to switch on. But only usable with DC.

 

just to add on LowQCab's suggestion... all those latching relays, inside are usually a DPDT relay with a simple wiring. You can make the exact same schematics if you dont have the latching relay availible in your country. For example, for me it is impossible to order from Digikey but it is really simple to make a self-latching relay with a standard DPDT:
View attachment 335673

Also check my post #80 on the previous page.

This requires two switches. I believe that @LowQCab was referring to a mechanical latching relay as in my post above his.

 

As far my knowledge goes latching relays also have two coils, set and reset.

 

As far my knowledge goes latching relays also have two coils, set and reset.
View attachment 335674

”Impulse relays” alternate states with a single input.

 

okay then I learned something new today too as mentioned, due to inavailbillity in certain countries, the same effect can be achieved with wiring of very commonly availible components.

 

We have not heard if the PB could be an SPDT button,r not.

 

I had a task to make an output toggle using only one switch, and this worked just fine:

Interesting circuit, although requiring four relays makes it a little expensive.

Below is the LTspice sim for your amusement:
It appears to operate similar to a master-slave flip-flop.
The plots show the relay coil currents, indicating when they are on.
Relay U3 toggles on the rising edge of the PB signal, and U2 on the falling edge.

 

If a quick press and a 2 second press would be acceptable, and the circuit is DC, then it can be done with one relay and an SPDT push button.