I can still vividly remember my first few lessons at pole classes. My instructor showed us a move or a spin and then asked us to perform it on the other side.
The groans that echoed through the studio are still repeated today. "I'm right handed, my left hand is useless" or "I just have no strength on that side."
It's a common reply to just shrug, give it a quick go and pretend that you just can't do it and move on, repeating the move or spin on the "best" side and smiling when you nail the move.
So, what's the problem? Well, a great example, would be myself. I am right handed. I write with my right hand, I use my PC mouse in my right hand and I naturally pick up objects in that hand. Imagine my surprise one day when I found out I was performing the Gemini on the other side to all my fellow "right handed friends" in class.
I had never noticed this until we attempted double Geminis in class one day and as I stood next to my usual pole partner, we realised we weren't "compatible". I couldn't believe it! I had never realised this discrepancy until now. The next few lessons I then discovered my capability to climb on different legs at the starting point and a few spins which were easier.
After discovering my almost ambidexterity it then became apparent that it came in fairly useful. Where we were shown moves where it was easier to invert on the opposite side or start climbing on the wrong leg, I was already a step ahead there, and didn't have to struggle with the first part of the move quite as much.
It wasn't that simple though and by no means increased my status to extremely advanced, it just became a touch more helpful. Switch it to the other side and it becomes a different story!
Of course you will still have a more dominant side (or a stronger side) - just the way you can write perfectly with one hand and the other looks like something a 3 year old could outperform on. I make it a part of my weekly training session to try the move on the other side. Sometimes, I surprise myself and it looks just as elegant, sometimes I fail completely and just can't quite get it, and very occasionally, I do it and it feels and looks ten times better.
The importance of training both sides isn't just so you avoid becoming a strange looking character with a large, muscley arm on one side and a rather skinny weak arm on the other, it's about tuning your body to be able to perform to a standard which means you can get in to any move and switch arms and legs and still feel safe and secure.