overcoming 'handedness'

I'm on an enforced break from the bike and jogging on advice from my physio - after I complained about a sore hip and limp after a jog she said I'd done quite a bit of damage and after some heavy manipulation on Friday, she said to rest for at least a week.
The problem, she said, is probably that I'm very right handed - she said that she suspects that on riding and running I am very right biased, so putting more strain on that side. Its come to the fore because I was running on tarmac (usual trails are a mudbath at the moment) and pushed a little too hard.
A bit of background: I'm 42, did very little sports up to about 10 years ago when I took up cycling more seriously - not competitive riding, just lots of road riding, touring, a bit of mountain biking. Last year I took up jogging as I thought I needed a more 'impact' sport to balance up the cycling, so I've been gradually building up - I now do a 5km run 3times a week to complement my cycling (yes, I know its not much, but I'm a very inexperienced runner, I seem to be built for the bike).
When I asked for solutions she suggested exercise ball work, pilates and yoga for balance - problem is, I've been doing this for about 5 years! I am reasonably supple as a result and I've a decent level of core strength, so thats not the main problem.
So, is there any way to overcome a right-sided bias in cycling/running?
The problem, she said, is probably that I'm very right handed - she said that she suspects that on riding and running I am very right biased, so putting more strain on that side. Its come to the fore because I was running on tarmac (usual trails are a mudbath at the moment) and pushed a little too hard.
A bit of background: I'm 42, did very little sports up to about 10 years ago when I took up cycling more seriously - not competitive riding, just lots of road riding, touring, a bit of mountain biking. Last year I took up jogging as I thought I needed a more 'impact' sport to balance up the cycling, so I've been gradually building up - I now do a 5km run 3times a week to complement my cycling (yes, I know its not much, but I'm a very inexperienced runner, I seem to be built for the bike).
When I asked for solutions she suggested exercise ball work, pilates and yoga for balance - problem is, I've been doing this for about 5 years! I am reasonably supple as a result and I've a decent level of core strength, so thats not the main problem.
So, is there any way to overcome a right-sided bias in cycling/running?
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brain surgery..?
seriously, it sounds like you need to choose between cycling or jogging unfortunately...
Do they actually mean that one side is stronger than the other? How did they measure and come to the diagnosis? I would be asking some probing questions and if answers aren't forthcoming seek another opinion.
Get a proper bike fit. Are your shoes/cleats positions appropriately for your body?
One legged riding is a waste of training time.
Anyway, I'm back again with her next Friday and I'll see if she can enlighten me a bit more.
I bought a trainer recently, some of the routines on the website have you pedal with one foot and the other either on a chair or behind on the trainer. Its rather unbalanced and as a result hard work, doubt it could be done without being clipped in.