BEWARE THE SWAY
If you try the head swivel, make sure you don't let your head sway to the right as your swing progresses. If that happens, the swing arc will change and you'll hit the ball inconsistently. I actually watch the clubhead go back, to see that it's moving correctly, but I use peripheral vision.
WHAT I THINK NOW
Your head also should not bob up and down. I sometimes would lift my head, but I eventually learned to keep it the same distance from the ground. That encourages solid contact.