Lately I've seen a great improvement in my handstand awareness and comfort by doing a couple of sets of this exercise, once or twice a week. It's obviously up to you exactly how and when you do it but other beginners like myself might find it helpful to include somehow.
Typically I might do something like this:
> Kick up, get stable
> Turn head and eyes to the left, look at fixed object - hold for a count
> Turn head and eyes back to the centre, hold
> Turn head and eyes to the right, look at fixed object - hold for a count
> Turn head and eyes back to the centre, hold
> Look through hands (if you normally look at the ground) - hold for a count.
You get the idea - lots of variations are possible. At first I would fall over as soon as I looked to the side, but now I can move my head, and eyes (separately or with my head) pretty much anywhere and stay up - it's really made my "normal" handstand seem 10x easier. Also, it's a good way to learn how to get your head straight between your arms - something that also would immediately end a handstand if I attempted it.
If you're a beginner and can't already do this then give it a shot, I bet it will help tremendously. Make sure you do come back to the centre before you end the handstand, to get into that good habit.