No this is not correct, it takes 2 legs to make 240 volts, but you do not add them together to get a higher amp rating, it is still only 20 amp service going to that panel, so 2-120 volt legs @ 20 amps each does not total 240 @ 40 amps, it equals 240 volts @ 20 amps

OK, got it
Each breaker is only supplying 20 amps per leg @ 120 volts each, if either leg should draw more than 20 amps the breaker will trip, BTW, are your breakers in the main 2 separate breakers or are they tied together?

Definitely tied together
If you have anything on the sub panel wired for 240 and the main breakers are not tied together if one side should overload and trip one of the breakers without tripping the other one you will have a very dangerous situation, you could damage the equipment, (it will be trying to run on just 120 volts instead of the 240 it is wired for), not to mention if a sub-panel has 240 supply going to it, and someone works on it and just shuts off one breaker then you still have one side "HOT".

Good explanation, always wondered why they were connected.
Equipment wired for 240 actually draws fewer amps than 120 does, and operates more efficiently.

Notice that 115 is listed first under voltage, and 14.6 is listed first under Amps, so wiring this pump for 230 uses half the amps.

Right, half the amps 'cause twice the volts. But to be fair that's not more efficient. If that was the case the 240 rating would be half THEN A LITTLE LESS than the 120 to show that it was using power more efficiently. Of course that assumes that the label tells the whole story, which I'm sure it doesn't
BTW, notice this says 115/230, this is the same as 110/220, or 120/240, they are all used interchangeably.

Tell me about it....always confusing to know which to use (110, 120, ...)
20 amps @ 120 and 20 amps @ 240 are the same, they are both only drawing 20 amps, just at different voltages.

Higher voltages comsume less power, that is why industrial equipment operates on 3 phase and much higher voltages, there are a bunch of different configurations in 3 phase, but the simple version is they have 3 legs, (Hot), each is rated for the same amperage, and if only one leg should trip then it does what is called single phasing, and will no longer work properly and most likely burn out.

Anytime you run wire, the farther you have to run power the heavier the wire has to be.