re: Browning LH bolt rifle...?

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Is he left-eye dominant? That's the determining factor, not whether he's left-handed. I'm right-handed, but left-eye dominant, so I shoot from my left shoulder.

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I am the same, right handed, left eye dominant, and I shoot from my left shoulder. The first bolt action I purchased for deer hunting was a left-handled action (Ruger M77 Mk 11) but for subsequent bolt-action rifles I’ve just purchased right handled actions. I can’t think of an instance in 25 years in which I required a quick follow up shot on a deer, which of course would be easier/quicker for me or any lefty with a left-handed action.

Not likely your son would need to rapid fire a bolt action rifle as the left handed sniper with the right handle action 1903 Springfield in the Saving Private Ryan scene posted above. You might be just fine purchasing the gun you want with a right-handed action, even though your son will shoot from his left shoulder. Just something to think about.