Hi Ben - I must apologise for feeding you some misinformation. When I saw the case back reference 980.006 I was on the quartz movement route.
You are obviously refering to the automatic movement and I've done some checking on my Divers. The details are as follows:

Ref: 844 has a Felsa FE 4611A (same as Jarls. See post January 2009 at classicheuers.blogspot.com)
Ref: 844/2 has ETA 2872
Ref: 844/3 could not be opened but will do so in the next week or so after a visit to the watchmaker.

I agree that it would be a great idea to get a definitive list on the movements. At present I am working with some of the other forum members on a reference table listing all the Heuer Divers. Hopefully it will be made public in the not too distant furture. The next step would be to detail the movements so watch this space.
Regarding your watch, it looks like it has had some dial work and relumed otherwise its in great shape. You mentioned the reference as 844. Is this correct i.e. just 844 or is there an additional number eg 844/2 or 844/3? It would help to know.
regards
Paul