I have a great LH Remington 700 BDL in 270, I also have a LH Browning Stainless Stalker in 7mm-08, and a custom LH MRC 308. The Remington and MRC have been tweeked with glass bedding and trigger work, the Remington sits in a nice laminated thumbhole stock, the MRC sits in a custom Clairo walnut stock. The Browning is as it came from the factory and even has the BOSS on the end of the barrel.
The Remington shoots MOA or less dayin and dayout. The MRC does the same, but looks a lot prettier.
The Browning having nothing done to it will match or out shoot either one, it's just a perfect rifle and was highjacked by #3 when he started shooting.
You may wish to look a Brownings, they offer LH in a number of calibers and are priced nicely. There's good and bad said about them, but mine is a gem and they sure do sell a lot of them so they must meet someones expectations.