Assuming that you start combat in range of the enemy with a Gaze and it is aware of you. You know that the creature has a gaze attack, either from previously encountering it or someone else making a knowledge check.
Can you do it against the initial saving throw at the start of your turn in Round 1, or do you have to wait and do it later in your turn, so you'll only be protected starting in rou8nd 2?
If you didn't have your eyes closed before combat (which is possible) or perhaps fixed on your feet (for averted) then you can't close/avert them until you get an action. I'm pretty sure the latter would be a hefty penalty on perception checks if allowed, going around with your eyes 'averted' out of combat isn't mentioned in the rules.
It does protect you (after your first round action and before your second round action) from the gazer using a standard action to force a second save.
That's what I was thinking, thanks for the reply!
Do keep in mind that averting your eyes only has a 50% chance of working (i.e. still a 50% chance to see meet the gaze attack)