"240v" air conditioning is usually set up for 240 v "single phase;" that is, two "hot" wires, each having 120 volts to ground, but measuring 240v between them. I suppose that, in theory, you might find a 3-phase, 240 volt air conditioner....but not very likely!
"277v" is the nominal voltage of one "hot" leg in a 3-phase 440v system. I have only seen 277v used for lighting, and never for HVAC equipment.
That things might be "wired wrong" is very possible. HVAC is an art of its' own, and I -even with several courses under my belt- am not qualified to comment on the specifics. I will say that you need a wiring diagram -either 'ladder logic' or a schematic- to do your troubleshooting. If one is not present, you'll have to find a HVAC person to look it over.
