The reason that it appears the outputs are ON as you said when the LED is off is because they are DC Sink outputs. That is, when they are logical 1 (ON) they SINK to ground. If you are testing output voltage to ground then you will see inverted logic (Outputs are internally pulled up to +V). In order for the outputs to respond in true logic, the + terminal of the powersupply should be connected to the + of the load and the ground or negative terminal of the load should be then connected to the output terminal. The output will be +V (24 or 12V) when off and 0 volts (approx.) when on.
By applying 0 volts (ground terminal) to the negative of the load (relay, solenoid, motor etc.) is just like putting a switch in the negative wire rather than the more conventional positive wire.
I hope this helps.
WireNut
