Someone may make a general statement that classical music is “relaxing” without referring to a specific piece or composer. Therefore, I would still contend that perception is not incorrect. Even the Firebird, which if you asked for a description of the emotions of this particular piece, would certainly not be relaxing. But the listener might find listening to be relaxing because it takes them away into another realm, far away from the daily world. I guess what I’m trying to say is that couldn’t a listener have a general observation about classical music, but then a different observation about a particular piece, and neither of them being incorrect?
I find classical music relaxing the same way I find reading a novel or watching a movie relaxing. Is the novel itself all about swans and moonlight and calmness? No, and it would pretty boring if it was. A poem, maybe not, but a novel or movie definitely.
Take for example a favorite movie of mine, the Hunger Games. A very intense ride, with a lot of very mentally-challenging themes and drama, in addition to the sheer physical chaos of the battles. But it's a change, an escape, a chance to enter a "different realm," as dogperson said. For a listener, it's an exciting journey--one that lets the listener temporarily forget his current situation and enter someone else's, a refreshing change. Even Stravinsky's Firebird is its own epic story, as are Shostakovich's Quartets and Symphonies.
That said ... you can immediately tell a musician in this way. Someone who has either played or can imagine himself playing these works has a very different experience from someone who doesn't. When a musician who has himself played Beethoven's Appassionata listens to that piece played in concert, he, who has himself dug into the piece, past what the audience sees, experiences it on a whole different level--not as watching a movie, but as living it. That is why when we as musicians listen to a great epic work being performed, we come to the end and feel as though we've lived through hell. A non-musician just sees it like a movie, as entertainment. And I've seen (and myself experienced) the shift that happens when a non-musician begins experiencing music in this way.
I hope this all makes sense.