Both ways are perfectly legit. In my opinion, one is better for boxing while the other is better for kickboxing.
My boxing coach insists that you punch with your palm down. This is more solid structurally and gives you a bit more bracing (and consequently, wallop) on short close-quarter hooks.
I'm not a fan of these for kickboxing simply because the battle is fought at a different range. When throwing a hook from within arms reach the angle at which your arm is bent will generally be 90 degrees or less. At this angle, a palm-down hook is SUPER solid and is perfect for the compact realm of boxing.
Once the arm is bent at an angle of greater than 90 degrees, however, the solidity of the structure is lost. Even worse, the thumb becomes exposed. One of my training partners dislocated his thumb by throwing a palm-down hook from too far.
It's for that reason that palm-facing hooks generally suit kickboxing better. Unlike their palm-down counterparts, these punches will align correctly when the arm is open at a wider angle. And even more importantly, the thumb is taken out of the dangerzone.
Experiment on a bag with different ranges and you'll see what I'm saying.
Highly opinionated but out of touch with the current MMA scene.
Support my FB athlete page!
Watch my 5th pro fight here
Watch my "Bare Knuckle" Boxing match here