Yes, always go up when the outside leg goes up, you will be counted off for this at shows. A good way to be sure if your on the right diagonal is to watch their outside "shoulder". That always helps me, good luck!
The idea of posting to the outside diagonal is taking your weight off the horse's inside leg while going around turns or in a circle.
When teaching students to tell diagonals by feel rather than looking, I would have them trot a very small circe (10 - 15 meters) first on the outside diagonal, then on the inside. You can absolutely feel that it's harder for the horse and more work if you're posting to the inside leg.
In other parts of the world the teach people to post off the inside diagonal. We had an exchange student from Puerto Rico who thought we were all crazy for posting on the outside diagonal. He was jumping Grand Prix level though, so we knew it wasn't that he just "forgot."
It only matters in classes like English pleasure and hunter type classes and in the US.
The is no correct diagonal in dressage and as mentioned earlier, posting on either diagonal is often done outside of the US. The reason I give to use the inside hind leg is on a advanced horse the the rider (assuming they are advanced enough and should be riding this level horse) following the natural rhythm of the post will have their inside leg is in the best postion for driving when the inner hind leg in its phase of support after its forward advance, so just as it touches the ground. This means the influence of the inner driving leg will impart a greater force and that hind leg acting on that force will push off with greater impulsion. This will now stimulate the outer hind leg to reach more aggressively forward in its swing.
So why do we want to influence the outer hind to swing more vigorously... This should become appearant in an arena with more close cramped quarters where turns ( not sharp ones) are commonplace. And that is because the outer hind leg MUST travel a greater distance on any turn than the inner hind leg. The rider's outside leg at the same time should act mainly as a support for the inner hind leg and requires less of a driving influence. So what happens is that the horse's outside leg is relieved of the load bearing and therefore becomes easier to swing farther forward to cover the greater distance it has to go.
Summery...The Europeans (mostly) are looking for better engagement.
all good information to know! I was 'raised' on posting when the inside front leg is down. It was drilled into my head and I can still hear my instructor saying 'WRONG DIAGONAL' or 'SIT ONE' When I moved down to florida and started taking lessons again, the instructor told me the opposite. I just kept thinking 'you're WRONG!' but, hey, they are the instructor right? They're supposed to know. In any case, I still post off the inside diagonal, as I was taught...and I find it fascinating that after all these years of NOT riding, and having trouble getting the right diagonal without looking back when I did ride...I actually pick up the right diagonal without looking automatically now!