you mean palm-down versus palm-up I take it? Efficient for what? Palm-up is more compound, palm-down is slightly more isolated to lats. If you want mass, compounds are efficient. If you are trying to use it as a back exercise, palm-down probably marginally better. Although truly isolating the lats involves taking the weak link (arms) out of the movement, like one of the many machine-style elbow-contact pullovers. very nebulous question. Probably your gym should spend more time lifting and less time debating mute points.
I do pull ups for back exercises, I have a good feel for isolating my back with them after doing them so much, rarely will I do a chin up for fear of overworking my bis.
I never really noticed too much of a difference with the two, but I'll give it a shot. I'm sure you guys are right that pullups target the back more. In the end, it shouldn't matter much. I agree with Maj. Tell everyone to lift. Arguing over minor things such as that isn't going to get anyone anywhere.
IIRC, palms facing uses one more muscle that palms away does not, hence they are easier. I do them for number goal so do them palms facing. But if you want to emphasize the main muscles it uses more (lats and bis), then do them facing away.
Variation is the key...do different types. IMO, you should work most on your weakest movement. Alot of people can do pull-ups easily and chins are tough (or vice-versa)...so work on the weaker of the two...but always vary it up.
You can also do them with palms facing (if your gym has that grip), or throw two towels over a horizontal bar, and just grab those and pull yourself up, this is good for your grip, arms, and back.
Chins (supinated grip) allow greater range of motion and biceps are in a strong position so you can work you lats harder because you are less limited by your biceps than you are with Pullups.
Pullups work biceps less because your hands aren't supinated but pronated, Pullups work forearms (like a reverse curl) and hit lats with a shorter range of motion.
I do both because I use pullups mostly for Forearms.