I have just seen a video on youtube and the amplified speaker has a speaker connection and it says feed to right speaker. So that says to me that the powered speaker is the left so when they are behind / at the side of me the right speaker (amplified ) is my left.
People have said not to worry what is left and right .I certainly wouldn't put the left cup on my right ear and vice versa and yes I do use headphones a lot for desktop use.
I don't think we will ever get to a right answer, as stereo setups just aren't designed for you to sit between the speakers. The best you can hope for is to find a sound you like.
Going back to what you said at the start, it sounds better when facing the speakers. were you in front of the speakers or kneeling on the sofa as if in the same position as when sat but facing the other way? If the former the improvement is sound is your position, not jut the direction you face. In the two quoted sections above, it sounds like you are treating the speakers like headphones, but they are two different things.
Imagine you are at a jazz concert. You face the stage, the trumpeter is on the left of stage. the piano on the right. Now turn your back on them. They are still in the same place. The trumpeter may be to your right, but he is still in the same place and you can still hear him in the same place.
Now imagine you have a really good live recording of that concert. You have a stereo set up perfectly and it has a really good soundstage, you are stood in the perfect position to listen. You can hear the trumpet on the left of stage and the piano on the right just like when you were there. Turn your back on the speakers. you will still hear the trumpet on the left of stage. There is no need to swap the speakers around. In fact it will sound bad if you did.
Now repeat the exercise with headphones. Whichever way you turn, the stage is still in front of you.
It sounds like you are trying to replicate the experience of headphones, or speakers in front of you while having the speakers to the side of you. It isn't going to happen, it will never sound right. It might be that the best compromise is to treat the speakers like headphones and have them the opposite way around as you are planning. I honestly don't know! The one think I would say with this regard about speakers and headphones being very different; when listening to headphones you are listening to one channel with each ear. with speakers you listen to both speakers with both ears at the same time.
It sounds from other comments that rearranging the room isn't an option. That's a shame as it could be the best option.