Effectively countering an enemy's unbalanced team
Solution 1:
In some cases yes you do want to create your own imbalance to respond to the enemy team's, but in other cases you want to stick to proper balance.
In all cases of a major class rush incoming, the Vaccinator is pretty good as it's likely that the majority of enemies will be outputting the same type of damage. Use fire resistance on a Pyro rush, explosives resistance on a Soldier or Demoman rush, and bullet resistance on the rest. While you do sometimes need a regular Über, the Vaccinator's superfast charge time makes it very useful.
You also need proper communication. It always helps.
The Scout rush is pretty effective if you let them get in; you need to kill them before they can reach you. High-level Sentries are the best at this, though if the Scouts are smart the Wrangler will be necessary to avoid Bonk Scouts. Protecting your Sentries until the Scouts go away will be paramount, since the Engineer probably won't be able to rebuild in the chaos. Depending on how much of the enemy team is Scouts, you might want to stop being Sniper if you can't get the headshots, and Spy is rather useless. Other than that you can beat a Scout rush with general combat classes. Of course the best solution to a Scout problem is probably stacking Nataschas.
The Soldier rush is probably uncounterable. Assuming they don't all have the same loadout you'll be seeing various types of rockets, some shotguns, a bit of mini-crits or crit immunity or healing, bombing runs, and Engineers' days ruined. Large quantities of Soldiers also tend to patch up their main weakness - you can dodge one rocket, but probably not five coming from different directions. There's not really much you can do about this even if you have a bunch of Pyros good at reflecting/Short Circuit users.
The Pyro rush will probably involve spastic flares at long range, so it's not completely worthless out there, but it will destroy you when they get in. It's not a terrible idea for some of your team to go Pyro to weaken the effect of the burns, but chances are most damage will be from the fire and not the burns. If you have a Sentry up it will do an okay job of keeing the Pyros away, but you should probably prefer to have mostly Heavies and Medics. Just keep shooting the Pyros and tossing each other Sandviches. Unless the Pyros are all Phlogging, don't bother trying to be a Soldier or Demoman (and if you do get a ton of Phlog Pyros, grab the Battallion's Backup). People who go Spy in this situation deserve what they get.
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The Demo rush can be one of two things:
The Demoman rush will be similar to a Soldier rush, but without the extra support abilities. Still plenty difficult to beat straight-up.
The Demoknight rush is kinda similar to a Scout rush in that they can't do anything until they're in your face. Try to avoid using explosives and fire on then and you should be fine.
The Heavy rush will win the game if you let it, especially on Payload. You will need Snipers, good Spies, and maybe Kritz - and enough combat classes to keep the Heavies from attacking this support. You must kill enough to avoid them coming together in a big clump. In the land of the Heavies the Teleporter is king, so be on the lookout for the lone Engineer.
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The Engineer rush depends on whether the enemy team is attacking or defending.
If attacking, it will involve copious amounts of Mini-Sentries and a few regular Sentries in addition to straight-up shotguns. Having a Sentry or two of your own will likely completely paralyze any further enemy progress unless they have the Wrangler and you don't.
If defending, it will involve craptons of level 3 Sentries and Dispensers. You will need a good Spy or two, some über Demomen, and team communication. If the entire enemy team is sitting back you can charge your übers without worry, so it wouldn't be a bad idea for the whole team to go Demo and Medic with maybe a Soldier or two mixed in and possibly a Heavy to kill the one or two Pyros the Engies hired for airblasting.
The Medic rush will involve chain übers - if it doesn't, you should be able to kill them easily. Stay out of their range until they're no longer invincible and attack. No special classes should be needed.
The Sniper rush is likely to involve more Huntsmen than rifles. As long as this holds true, you should be able to kill the Huntsmen without too much trouble and then deal with the back line. If they're all back-line riflers though, you'll probably die before you can hit them. Common knowledge is to send in a Spy or two. Of course the Snipers will be expecting this, but sometimes it's all you can do if all the Snipers can aim well and are covering the objective you need to capture. This is all assuming you don't just use an Übercharge to walk up to them and wipe them away, which can be possible if your über target is fast enough to get there in time.
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The Spy rush can be one of two things:
Covert. You see no enemy team for a while and then everyone's stabbed. Best strategy here is to be suspicious when it's been too quiet, or you see a Scout rush that gets wiped out too easily. Having a smart Pyro or two also helps.
Obvious. The Spies will just shoot everyone. You shouldn't need any specific classes to deal with this as long as you can focus one guy at a time. Pyros aren't actually that good in this situation; they only counter Spies that are trying to stay hidden, not battlespies.