IMO tubes can be very clean, absolutely. That doesn't mean they have no apparent harmonic distortion (the pleasing kind), it just means their presentation is clean and not biased to any particular part of the frequency range.
Given the way tubes work I don't think it would be possible for them to be dead-exact to SS. I think it's a more fair argument to argue if tube distortion is a good thing or bad thing (it's less perfect and therefore bad, it's more pleasing to the ear and therefore good) than it is to argue that tubes don't change the sound.
Also, with different tubes (tube rolling) it's perfectly sensible to hear differences from different manufacturers. With different specifications, different material choices, placements, filters, etc, many would not apply gain (or the same amount of gain!) the same way. Not all will do so linearly across the frequencies. Many vintage (and even current) tubes were designed for military use in various situations. For those tubes some are voiced differently to bias the frequency that will be its dominant special use, or to bias against its likely background environment, etc.
I also think the tubes have greater effect in some headphones than others. I'm not sure they have the same effect on low impedence, high current phones like AKG 70x as they do on moderate-high impedance headphones like Senn HD650, HD800, Beyer DT880.
My own personal feeling on why tube sound can be pleasing resulting from its distortion is because, to me, there is such a thing as "too perfect" a sound. All the data-mining detail cravers out there will disagree, but I think SS can be very sterile in a way that, while it's more perfect to the recording it may not be perfect to the way real music would be heard in a real room. The room itself changes the sound before it gets to your ears when you're hearing real music. The microphones in the recording studio on most recordings try to avoid recording the room (excluding classical which explicitly records the room, and excluding binaural recording.) The idea is your room speakers can re-create the room. In that sense, SS is fine for speakers since your properly treated listening room will recreate that natural distortion between you and the speaker. But headphones don't have that luxury, the sound is being beamed into your skull directly with little air, and no room distorting the sound. There I think the tubes recreate a little of that natural air/room distortion in the analogue space in a way that is very hard to recreate digitally. I think tubes are more important in headphone land than in speaker land for that reason. Purely my line of thinking, this isn't the sound science forum after all!