Initially, I was going to write about whether classical music can be offensive, phrasing it as a question. I got a few sentences in, and then I came to understand that the answer is, of course, yes. Offensiveness, like art, is meaningless without consideration of a viewer. The original phrasing of the question invites a poor understanding of the issue raised. This understood, we revisit the question more honestly as "what is the nature of offensivity, and how may it be understood/realised in terms of music". That's actually three questions, some of them possibly related to questions of science, some to questions of value, some wrangling over definitions, and some questions of data. Deep philosophical understandings of things comes from understanding and being able to navigate between these things, and in having a sufficient mastery of language to not be easily swayed by initial phrasings of concepts or questions. Deep understanding is a translation between ideas in one person's framework and ideas in another. Without a strong philosophical identity of one's own, one is a child waiting to be tricked.
Back to the question at hand, I think one kind of offense which would be particularly easy to realise with music would be to demean a known melody or theme that one identifies with. There are some sounds that are almost universally seen as humourous (bodily noises), and some variations that are seen as being less formal (polka). Thus, if we took a song that someone saw as part of who they are (e.g. a national or ethnic anthem, or perhaps looser associations like songs one particularly likes or those that have other identifying elements), and inserted such elements (Spike Jones' "Der Fuehrer's Face" and "Bohemian Polka" are good examples), it may be offensive to such people. I might speculate that people who are not offended either do not have sufficiently strong ties to these things, or that they are sufficiently prone to analysis that weakens the emotional aspect of offense (this might suggest that cynicism and analysis are two paths from the emotional, honour-centric ways some cultures passed in the past and some are in presently. If we look at it this way, it is disappointing that the first seems to be more prevalent.. but then, it's the easy path). This kind of sharing of identity can take place in many shape and strengths, from a person who sees themself as better than others because classical music is part of their upper-class heritage to an ordinary music snob to a patriot of some kind who sees a song and the culture it is tied to as part of a cohesive identity they love.
I must confess a certain amount of protectiveness for a few songs I feel about this way, although I have a certain amount of cynicism and a lot of tendency for analysis that makes me less prone to offence. When I was younger I had a fanciful notion that I was not offendible in such way, perhaps partly fueled by the fact that people only rarely confronted people like me with things that might offend (it's more difficult, because I don't identify at all with the United States). It is seen, as part of what I'm tempted to incorporate into the "liberal culture" of the United States, that most concerns about honour are outmoded, that violence used to support it is barbaric, and that most concerns of people being offended are their responsibility, not that of the offender. This is, of course, a societal decision, and one that's not uncontested by more conservative/traditional components of our society. I am concerned that the liberal factions, which I loosely identify with, are effectively nihilistic (by consensus) and don't represent an alternative order (except perhaps societal destruction a la libertarianism and other radically individualistic ends). More relevant here, I am concerned that the liberal factions are opposing parts of human nature without portraying this as a struggle. Without that acknowledgement, people will not design institutions that are realistic for people to take part in with integrity, and people will not recognise their internal struggles or feel accomplished for overcoming the more harmful kinds of individualism.