You're going to do the harder part with your left hand anyway
I have been teaching guitar lessons for a long time. I've taught every type of student you can imagine-- young kids who can barely count or know the alphabet, retirees who are seeking a hobby now that they have free time, people who gave it up earlier in life and are giving it another shot, seasoned professionals looking to pull themselves out of a rut; you name it.
One of the most common questions I get from absolute beginners who are also southpaws is 'shouldn't I play lefty, since I'm left handed?'
My answer is always a resounding 'no', and here's why:
1.) Left handed guitars are more expensive
All of the machinery and tooling used to route the body shape and headstocks of these instruments have to be flipped around backwards to make a left handed guitar. What that amounts to is more time; and time, as we all know, is money.
An extra 50 bucks for an entry level guitar doesn't sound too bad, so if you'll feel more comfortable, what's the big deal? Unfortunately, the higher cost increases geometrically with the cost of the instrument as well. Meaning, a $150 guitar might have a $200 lefty version, but a $3,000 guitar might be an extra $300. As quality increases, so does the time invested in building an instrument. Therefore, the time to create one that's backwards from what the factory is churning out on a regular basis is greater. Given the general lack of left handed guitarists, a left handed instrument will also be more difficult to sell than its opposite counterpart. And what about checking out your buddy's new guitar?
2.) You won't be able to play your friend's guitars
Your best guitar bud just dialed up a sweet new SG Custom-- 3 pickups, alpine white AND gold hardware?? Well guess what? You're not going to be able to play it. And not because he's a selfish little prick either. You're going to flip it over, and everything is going to be upside down and backwards.
You'll likely always have to rely on bringing your own guitar to friends houses, on camping trips, or to jam sessions. In the event you're performing and there's some sort of technical snafu with your axe, that super solid dude you were talking to before your set *would* loan you his guitar... But you can't play it.
Speaking of playing things:
3.) The hard stuff is all done with the left hand anyway
The teacher I studied with in Boston for over 10 years, as much of a mentor as he was a teacher, and the most amazing guitarist I've ever seen, was left handed. And he played righty. Two of my coworkers at the spot I taught at up there for ten years were lefties who played right handed, and my guitar buddy from when I was a kid was a lefty who played right handed as well. B.B. King and Duane Allman also preferred the sinister metacarpus for things other than playing guitar.
One of the precursors to the guitar was a Greek instrument called the lyre. The more challenging part of playing the lyre was done with the right hand, but as the instrument evolved, the orientation did not change. So you will be ahead of the game when you're using your more dominant hand to find the right note, on the right string, on the right fret, with the right finger. The right hand just goes up and down, at worst finding one out of six strings. A 24 fret guitar has 118 different possible places to put the left hand, as well as deciding which of the 4 fingers should be used to play the note.
This final point is usually what sells people on playing 'right handed' when they are in fact, a lefty:
4.) There's no left handed piano
It doesn't exist. You just do one thing with your left hand (the bass part), and you do the other thing (the treble part) with your right. Furthermore, a left handed saxophone did not exist until 2004, and it was a custom build for a player who had suffered a stroke. Trumpet, trombone, and accordion don't have left handed versions, either. You just do one thing with one hand and another thing with the other; and a left handed violin is extremely uncommon.
It seems like the guitar, electric bass, and drums are the three most common instruments that are played left handed. Perhaps it's because of the large number of people who are self taught on those instruments, and aren't made aware of the benefits (for the guitar, at least, I'm not a drummer) of playing with the standard orientation.
All in all, if you're seriously considering the long term investment of learning how to play the guitar; playing with a right handed orientation will likely make your life easier in the long run.
On the other hand, if it feels as wrong as a three dollar bill, then go with what your instincts tell you.