To some Tai Chi people it’s important to know where the eyes are looking when doing a Tai Chi form for slightly esoteric reasons: “your eyes lead your intention and your intention leads your chi.” But I think we can come up with reasons for using the eyes in Tai Chi that require no mention of intention (yi) or chi.
Try this: Sit comfortably. Turn your head slowly to the side, back to the middle then the other side. It’s a typical neck stretching exercise that you’ll find done at the start of your typical Kung Fu, Tai Chi, or Yoga class. It’s good for your neck, but there’s nothing particularly special about it.
Now try this: Instead of just turning your head to the side, actively look to the side. Lead the movement with your eyes looking to the side. Now compare the feeling of doing that to the feeling of just turning your head to the side.
If you’re anything like me you’ll find the experience quite different. When you are looking to the side for a reason your whole body co-ordinates better, not to mention, I think you can turn your head a bit further too. I’m sure there is a scientific word for this purpose driven movement, but I don’t know it.
Think of an owl, when it turns its head to look at something interesting that might potentially be prey – the eyes are always locked in.
When doing the Tai Chi form, try actively looking with your eyes and turning your head in the direction you are going. Hopefully you’ll notice the different in the quality and coordination of your overall movement.
The Tai Chi Notebook 
This is a great exercise. Thanks for bringing up this topic. We don’t talk enough about training the eyes or head rotation. When I was doing postural bodywork, I taught that same exercise. The only other cue I would give was to keep the eyes on a level plane as they led. You want your head to stay on axis just turning. People have a tendency to look back and down over their shoulder, or to move the whole head and curve the neck around. Just turn the head and keep it on axis.
If your neck is stiff turning to one direction, turn to the easier direction. Stop at the end, and take a few full breaths. Repeat a couple of times. Turn to the harder side and see if has loosened up. Usually, doing this will start balancing the neck movement. Never force it in either direction, which will tighten up the muscles.
I’ve also found it helpful to touch the neck muscles with my fingers and explore what is tight and sore as I turn and breathe. Usually, this awareness alone will help the muscles relax, unwind and balance.
“Pivoting” the topic, one of the big changes I had to make as I became familiar with Hong Junsheng’s teaching method was how to use the eyes. Previously, I had been taught to focus on a hand or finger and follow it as I did the form. Hong had students focus on one point at a time as they did the form and push hands. That is, you focus on one point external to the body during one section of the form, then as the form changes, you change your “target” as you proceed through the form. Push hands is harder to explain in text
The reason is to fix your head in space. Like your owl photo above. If you were to pick up an owl, or one of several other birds, used to being handled, and moved their body around, their head would remain in one place, and at one attitude, until they changed the focus of their eyes. This is a desirable trait for us in our training.
This is related to your original exercise but inverse.
To see what this does, find something in which you can see your reflection, a mirror, glass door or window, etc. Watch the top of your head of your reflection but let your head turn as you turn your body. Usually, the top of the head will wobble around.
Next, focus on the top of your head, turn your body without allowing the head to turn. Usually, the top of the head will stay dead still. It will start taking on the characteristics of the suspended head of the bird.
The usage of this in taijiquan is, of course, so you can learn to fix an axis in your torso to give your movements rotational leverage. Like the turning of the head exercise, it also loosens the neck muscles and raises your kinesthetic awareness.
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