"Some amps sound warm and involving...some sound cool and distant. The same can be said about speakers. And most folks don't dispute this...generally. Can the same be said be said about a table / arm combo?"
John, turntables are much different than CDPs, preamps, amps, and even speakers in that they take on very different sonic personalities depending on how they are configured. Everything, and I mean everything, changes the sound, sometimes drastically. That is why VPI offers so (too?) many tuning options via their accessories.
The sound of a VPI table/arm combo can be made to lean toward cool and analytical or warm and musical depending on how it is setup (suspension type, platter type, armboard, base platform material/construction, etc).
Regarding a VPI "house sound": Just as Stanwal points out, the older suspended VPI's (HW-19/TNT) tend to sound ricker and darker ("a slightly warm, embellished and romantic sound") and the newer ones with spikes and rigid suspension tend to sound more detailed and faster (accurate and incisive). The cool thing is that you can vary the sound of any of them quite a bit by changing their setup.
I hope that you find this post more helpful than my last one.
Good luck.