has anyone done a reputable study on it?

There are a lot of studies, with murky results. FWIW, the original context described by Bandler and Grinder is that when they stood on stage and asked audiences questions, they noticed that huge portions of the audience would make the same eye/head movements in response to the question.

As far as I know, none of the studies performed such a test. ;-)

However, I've also seen a book which discussed how certain head and eye positions affect blood flow in the brain, suggesting that tilting the head back and to either side directs more blood to the visual cortex in one hemisphere or the other. And it's plausible that the eye movement is a precursor to that movement -- I notice that when I start to visualize, my eyes go up first, then my head.

Anyway, the original NLP-based generalizations were not terribly accurate, and even the guys who originated them don't consider them to be of much importance any more. I've rarely bothered to use them in my work, since I can't see people's eyes over the telephone. If you're a good listener, you can identify someone's processing mode by sound almost as easily as you can by watching eye/head movements, on the rare occasion that you need to know.

(Note that people's head tilts are usually accompanied by postural shifts that in turn affect voice timbre... which is also how we can identify many emotions expressed in voice tone -- the postural shifts and muscle tension differences show up in the sound.)