Honestly it depends what you're doing. A tag name is only correct when it unambiguously specifies the content. As in, a header is just a header - there's no "dual nature" that we have to support by allowing multiple tag names. This means the ontology you're representing has to be "linear" - you can have more specific and less specific tags, but never a "fork" where you could really choose either more-specific tag in order to emphasise a different "part" of the tag. Classes should always be for secondary properties, which are not the main "essence" of the content.

All I say above is a just a particular way of thinking about document markup, of course. If you don't agree with it, then tags are probably the wrong way to express what you're thinking of.