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Hi Joe, I'm Brian and I'm here to help!
It sounds like you have a heat pump and this is a very common question. The O terminal and Orange wire is for your heat pump's reversing valve to switch it into cool mode when the thermostat is set for cooling. The reversing valve fails to heat mode so it must be energized in cooling. The Honeywell RTH6580 thermostat should have two rows of designators for the terminals. One row should be for 'Conventional' wiring and the other row should be labeled 'Heat Pump'. You should be using the Heat Pump designators. The terminal the orange wire should connect to will be marked O/B in the heat pump designator. The thermostat is preprogrammed to operate that terminal to energize the terminal while in cooling. It can be changed for a Rheem heat pump to energize in the heating mode but unless you have a Rheem heat pump outdoor unit, there is no need to change the operation of the terminal.
Once the thermostat is wired and powered up, you need to go into installer setup and tell the thermostat you have a heat pump system either with or without emergency backup heat (whichever is your case). If you had wire(s) connected to E or Aux or W2 on your old thermostat, then you should choose 'Heat pump WITH auxiliary heat' for the installer setup system type.
Let me know how that works and if I can help further.
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