Although I suspect GuyBetsy can tell you more, here's what I recall:

1) Cathay Pacific has a minority stake, with the majority controlled indirectly by the Chinese government (the Chinese government also has a minority stake in Cathay Pacific).

2) At the time of Dragonair's formation, it was given the exclusive Hong Kong carrier-based right to fly to mainland China. Cathay Pacific had to cede some routes to Dragonair.

3) Hong Kong aviation policy (in line with its tendency to favor cartels), allows only one HKG based carrier to fly to a given destination. In the past, Dragonair tended to get the lighter traffic routes.

4) CX and KA have joint fares on many routes, although KA has recently expressed its intention to take CX head on (KA has several larger aircraft on order).

That's all I know and some of what I don't. Hopefully GuyBetsy can correct the inaccuracies.