I have just had 13pin dedicated electrics fitted by Mitsubishi. I plug my old year 2000 trailer tent in, using an converting adapter which I know works fine. But I get the reverse and fog lights round the wrong way. Mitsubishi plugged in there light board and all works well.
Either your trailer, your adaptor, or Mitsubushi have got them connected the wrong way round. Check the Mitsibushi socket first as that's easiest. Here is how it should be wired :
To check it get a cheap multimeter, or even a 12v bulb with wires attached. Poke one test wire into connection 3 as the earth, and the other into connections 2 and 8 in turn with the car reverse lights and then rear fogs on in turn. If they are the correct way round, you trailer or adaptor are wired wrongly; if they are wrong than Mitsubushi got it wrong, though that's less likely.