I tend to agree with Remmer.
I ride both english and western (never "gamed" however) and have always used the same cues for both and never had a trained horse (either discipline) misunderstand them unless *I* gave them wrong.
First thing I would want to do is rule out pain. Arthritis or hock/stifle issues could easily cause a horse to refuse to pick up a canter or refuse to take a lead. That she picks up the right lead easily but never the left tells me that she's trained and understands but for one reason or another can't or won't not that she was never trained to understand leads (I can't imagine a good gymkana horse/rider not caring about it as leads can be essential in some of the gaming classes!).
Have a vet check her. See if she'll canter on the correct lead on the lunge, without tack, with tack, with tack and an experienced rider. If she'll canter with no tack, but not with the saddle on then it could be a saddle fit issue. Same if she'll canter with no tack or saddle and no rider, but won't canter with the rider's weight in the saddle (I said experienced rider here because if the rider leans or is unbalanced at all this can cause lead issues as well).
Next thing is to rule out you. Again, will she canter untacked on the lunge, what about you on her on a lunge (could be an issue with you leaning and blocking that shoulder or something else)? Will she canter for an experienced rider on the lunge? off the lunge (could be a signal you are inadvertently not cuing quite right or it could be she has your number and the left is her weak side)?
Once you've got all of that ruled out and she's happily cantering on the lunge line in both leads then I'd say you can assume you need to train/retrain leads. I'd have a trusted trainer help you with this if at all possible; it will make it quicker and less frustrating for all involved. An experienced trainer can better feel *just* when to cue a horse to make it more likely they take a certain lead, know how to open the right lead off shoulder up and close the wrong shoulder, know how to bend the horse, etc all to make the right lead easy and clarify what they want out of their signal the first few times rather than someone like you or me: canter, nope wrong, stop, canter, nope wrong, stop, canter, nope wrong, stop, etc. ad nosium (been there). They can also help teach you how to cue and set up for the canter especially for that "difficult" lead and can probably show you some exercises (even at different gates) to strengthen her in that direction.