How to pick a tubed integrated without hearing one

As other comments suggest, the efficiency of your speaker will play a big role here. I'd choose a speaker first (keeping in mind that you want ones that pair well with tubes). For speakers, you will want to think about whether you want very high efficiency ones that play well with SETs and what are known as fleawatt amps (a great budget example is the Miniwatt N3 or the S1), or a speaker that needs the 10-50 or so watts that a push-pull design can give. Most bookshelf and monitor speakers are of the latter type. Many in this design have quite low efficiency.

Hmmm. First question I would have asked-

1. What are you listening to now?

2. Do you like that type of sound?

I have found that even if you go to Best Buy and A/B cheap receivers against each other. ( easy to do with their cheap wired all together demo system ) you will notice differences in sound, Some with better midrange, some with better bass, some so bright they make my ears bleed, and some just blah. But still each receiver brand has a signature.

So if you lean toward a warm sound, then that sends you in a certain direction with tubes. EL34-6L6-5881 more warm with euphoric midrange. KT88 a bit warm with more bass. 6550 I find the cooler of the any mentioned here but with good bass.

Some of what folks have mentioned here are not even integrated. Also some mentioned intergrateds are well over $1000 used. People get carried away and want to mention their favorite and forget the question. In your price range, you will be limited with your choices. If your lucky, used CJ, Primaluna, Cayin, Sophia ,Consonance, Jolida or some off Chinese brands. New, you might try Jolida.

IMHO, auditioning "equipment" makes for a fun afternoon, but in all reality it's a waste of time - if the piece is not in your listening space and combined with the rest of your system. When you go listen to a system, what are you (predominately) hearing that you like / don't like? The power? The physical placement? The teleportation tweak? You just don't know.

If you spend enough time on this site (along with other sites) you'll start to see common pairings.