On my trips to Taiwan, I almost always flew Cathay Pacific, which is based in Hong Kong and is a OneWorld Alliance member. When that was not available, I would fly Eva Air, a Taiwan carrier. I never flew China Airlines, the flagship carrier of Taiwan (the whole political naming issue often gets China Airlines confused with Air China).
Why? China Airlines has a terrible safety record. A long history of pilot error and poor maintenance makes it one of the more dangerous major air carriers. An explosion in Okinawa a few weeks ago is just another blemish on its record. Just after pulling in to the gate area, the plane exploded in a ball of fire.
Not wanting to further damage the reputation of the airline, China Airlines proceeded to paint over the logo on the plane as it awaited cleanup. I guess they didn't want ignorant passengers who hadn't heard of the story to see the plane in person?