I am also a lefty shooter and have spent the better part of my life hunting with lever guns till the last decade or so when left handed rifles became available. I now have a fairly good selection of left handed rifles in my safe....here's my 2 cents worth...

Remington 700 BDL...they are pretty easily found used for a reasonable price (can be had for $500 all day long if you look)....great rifle that will last you a lifetime and easily found in 270...also in 30-06, 300RUM and 7mm Mag....I have.....4....including a 270 that I have taken piles of deer with over the years (maybe as many a 100)....one of the most accurate rifles I have...only thing I didn't/don't like about it was the factory trigger, but a good gunsmith and $50 will turn that factory trigger into a sweet trigger.

Tikka T3 Lite (Stainless)...mine is in 270 WSM and currnently my goto whitetail gun...I can't say enough good things about this gun...man it's acurate right out of the box, bolt is butter smooth, trigger is very nice and easily adjusted and breaks like glass, detachable magazine...handles and shoots sweet....can be found for probably $650 new i am guessing.

Browning A-Bolt....mines a Madallion and is in 25WSSM. It too is a great gun, handles very nice, detachable mag, easilly adjustable trigger that is nice (not as nice as the Tikka mind you), great ballance...haven't hunted with it yet, but it pnches paper sub-MOA all day as long as I do my job...one in 270 can be found easy enough for under $600 (non-Madallion).

Savage 11 Trophy Hunter XP....mine is in 7mm/08....another great cailiber worth considering for whitetail...low recoil, very accurate and advantage of high BC 7mm bullets. Mine has the accu-trigger (very nice, easily adjusted...almost as nice as the Tikka), detachable magazine and is very accurate, light and handles very nice. Can be had new for under $400 if you look...and with the package scope they are closer to $500...and the scope will get you buy for most hunting needs.

Savage 10....mine is a tactical model (FLCP-K) in 308...not really a hunting rifle, but it says volumes about the model 10's....again, very acurate, adjustable triggers, a little more $....500 - 600 for a non-tactical model.

I have others, but those are the 4 left handed guns I would look for...if yer gonna buy a Remington, buy an older used BDL in my opinion...hard to beat that rifle. My 270 and 7mm have both taken tons of animals in several different states and have always served me well. If I were gonna own just one left handed rifle it would be my Rem 700 BDL in 270...but if I was going to buy new, I'd have to say the Tikka has a slight edge over the Savages in my experience....just a slightly better rifle.

Can't go wrong with a 270, it's a great caliber and versitile and is a great whitetail caliber.