Am I going to have trouble flying with a reversed name on the ticket?

In my experience, they never care if it's not exactly right. I have a difficult name, and I stopped worrying about errors a long time ago, because travel agents never get it right, and their system will mess up the formatting anyway.

The rules say that the name on the ticket must be exactly as in the passport, but I never follow that rule. The only time I've been in trouble, was the time somebody at work ordered a ticket for me in my full name (despite my specific orders not to). Since the name is long, there wasn't room on the boardong card for my last name, and all characters outside the standard English 26 character alphabet became garbled.

I very much doubt that, with the thousands of passengers they see every day, I'm the only one who shows up with a "non-accurate" name.

One example could be a passenger from, let's say, Russia or Japan, where names are often reversed.

If the travel agent cannot change the ticket, you've done all you can. If there's a note attached to the ticket (I've never heard of this procedure), and the boarding card, which is the document you will actually need, is correct, I can't see a problem at all.

Other answers say it all, but I will just add a personal experience.

My name is Mohammed, do you know how many variations my name's spelling has?

Mohamed 
Mohammad
Mohamad
Muhammed
Muhammad
Muhamad
Muhamed

Basically, every boarding pass I ever had was with a different spelling, and I never had any problem. In your case spelling is right, no one will notice. Just enjoy your flight with your girlfriend Jones ;)

I had a similar problem where my hyphenated last name (example: Rosa Saragoza-Mendez) was accidentally reversed on my passport so that it appeared like Rosa Mendez-Saragoza (not my real name). I traveled internationally a few times while using this passport, and it was never an issue. Granted, that situation was different than having two documents with similar, but reversed, names.

On some occasions where the airline ticket vendor did not require me to use my name exactly as it appeared on my passport, I would use my correct name instead of the version printed on the passport. So then my ticket would have my real name, and my passport would have my reversed name on it. Since it was an obvious error, I would just explain to the people at the ticket counter that my last name had been accidentally reversed. In my experience, it was never a problem.

I often seem to travel with "Mr" appended to my first name, because my travel agent isn't very good at distinguishing between the name and the prefix. I've also had my name switched before as you describe. No-one (airline check-in, security staff, etc.) has ever even mentioned it. I highly doubt this is going to cause an issue; it's an obvious mistake. Relax and enjoy your holiday :)

For those who are looking for a more recent answer, I am happy to share my experience of July 2018. I also mistakenly switched my first with my last name when booking my flights. I was not able to change it since I booked through a travel website who were unhelpful (mytrip.com!) and airlines absolutely refused to assist since I didn't book directly. I then gave up trying to fix it and just decided to chance it at the airport. I was flying from Israel to USA with a stopover. When checking in they couldnt find my booking right away, as they always scan your passport and in this case it didn't recognise my name. I had my booking nr ready for this situation and insisted on them checking manually in the system. I still don't know if they noticed the mistake in name or not but they issued the boarding passes - still with wrong name. Once you are through with that it is highly unlikely that any official or machines that scan your boarding pass will notice the issue. On the way out from NY was more complicated as it didn't go unnoticed and took some time to issue my boarding pass but the impression i got was that this is not considered as severe as a spelling mistake. At this point she was not able to change my name and check in lady walked me to passport control in case there was an issue. Luckily it went unnoticed and from there it was smooth sailing. At my stopover in London I again mentioned to an official that my names are switched round as the self service machines would not let me through but again, no issue was made. So I hope this can calm you down abit and no need to spend the absurd fees some agencies/airlines try to squeeze out of you. Safe travels!