Remappable/reconfigurable controls has been one of the more successful calls for game accessibility in recent times, helped by the likes of Chuck Bittner and other advocates (examples: 1, 2, 3). There's such a broad range of people using idiosyncratic ways of playing, often with non-standard controllers, that you'd think the vast majority of game designers would keep this in mind.
So does that mean that most games now have the facility to set-up controls to suit your own playing style? No. Not yet, not even for simple left-hand play modes. In general, things are good on PCs, and bad on consoles. Meanwhile, there's a few things that can help:
There are a very small number of off the shelf Joypads that allow you to remap buttons, such as the Thrustmaster 3 in 1.
Some gaming devices allow you to move interchangable control pod clusters such as MadCatz MLG Pro Circuit controllers, eDimensional's Access Controller and the hard to find Radica Phoenix Revolution controller.
Switch interfaces allow for controls to placed within easier reach, in a highly versatile but often quite expensive way.
Some controller adapters, such as the Max Shooter allow a keyboard to be used to control game functions, and to be configured as needed.
Re-mapping modules from the likes of Chinese boffins, XCM, enable PS3 or Xbox 360 reconfiguration (to a degree). See a short review and video over at SpecialEffect's GameBase of XCM's Xbox Remapper and PS3 Cross Battle Adapter.
So does all of this weaken the call for reconfigurable controls in games? No, not at all. Being able to auto-load and auto-save favourite profiles within games adds comfort and convenience for all. For many, it's the difference between a playable and unplayable game. Seems like a small thing to ask to me.