The rising sun is in fact a symbol from P&O (parent of Princess Cruises). It is actually the top part of P&O's coat of arms. The official coat of arms is four symbols (the lion of Britian, the dragon of the Orient, the kangaroo of Australia, and the elephant of India) set on each quadrant of the P&O houseflag (whose colors are from Spain and Portugal, the P (for "Peninsular" of P&O)).
The coat of arms is on display on a stairway landing on Pacific Princess. I was looking at it one night on the way back from dinner - I got back to my cabin, and the next day's Princess Patter had a history of the P&O coat of arms (so I didn't really know all this myself
I would assume the coat of arms is on display on the other ships of the fleet. That and the portriat of Her Majesty the Queen are probably standard gear on these British ships...P&O has a stong sense of tradition, so I would think they would have these two items on each ship.