Welcome to FT!
Last July I took a ten-segment trip that involved departures from the U.S., Japan, Singapore, and India (from Chennai, where you'll be, and Mumbai). My name was messed up worse than yours: aside from the two being reversed, my middle initial was tacked onto the beginning of my family name (which was entered as my given name). In other words, if I were Thomas A. Edison, my name was entered as Aedison Thomas.
I encountered no security or boarding problems at. The only two issues were:
1. Check-in agents couldn't look up my reservation in the normal way. When I gave them the record locator or suggested entering my given name as the family name, they found it right away.
2. I didn't get automatic frequent flyer mileage credit because the name/number check failed. I had to follow up for retroactive credit. At the end of the day I got all I should have. Fortunately, I didn't need those miles right away because there was a delay of a few weeks getting it all straightened out.
I would not worry.