I have two Yongnuo YN-568EX-II speedlites. I can use them as radio/receivers between the two, but I want to use a controller on camera to be able to use the speedlights off camera without losing E-TTL and HSS.
Which controller(s) allow for off camera placement while maintaining E-TTL and HSS?
I'm currently thinking of the Yongnuo YN-622C-TX—but is this compatible? Is there a compatibility list?
Those don't have radio receivers built in, you need the 600 series for that. With these, you will need to add the YN-622C transceivers for the flashes, and either another one on the camera as a transmitter or (preferably) the controller you already mentioned.
It is much easier to use the controller (622C-TX) on camera than another 622C and the camera menu.
Edit: Changed 622N to 622C due to confusion between what the OP said in the question versus the Canon tag. I am assuming at this point that the OP is using a Canon camera.
The Yongnuo YN568EX II flashes have built in optical receivers that operate on Canons optical wireless system that uses very short pulses of light to communicate. You need an optical Master flash (such as another YN-568EX II) or Canon compatible wireless optical controller (such as the YN-ST-E2) on the camera to control them wirelessly without attaching additional radio receivers to the flashes.
You also need line of sight between the commander and the receiver on the remote flash. This works pretty well in dark studios or even moderately bright indoors areas. It is much more limited outside in bright light that interferes with the optical communication protocols of optical systems that were used by Canon and Nikon for many years. There are other limitations of optical communication compared to radio communication as well.
The advent of reliable and cheap third party radio triggers finally forced Canon and Nikon to transition to radio communication with their latest flash systems. (It also made Pocket Wizard, who make expensive and unreliable radio flash controllers, irrelevant.) Canon introduced their first RT flash, the 600EX-RT, in 2012. Nikon didn't follow suit with their first radio controlled flash, the SB-5000, until 2016.
The best way to control your two YN-568EX II flashes off camera with full HSS and E-TTL capability is to place a YN-622C II receiver on the foot of each flash and use a YN622C-TX transmitter on your Canon camera's hot shoe. If you later decide to add another flash, the YN685 has a built-in radio receiver that is compatible with the YN622C-TX. (But the YN685 has no optical wireless capability at all.)
If using one YN568EX II on the camera to optically control the other off camera flash has worked well so far in the situations in which you shoot, a cheaper option would be to use a Yongnuo YN-ST-E2 if you can still find one for sale somewhere (pretty much everyone has moved on to radio triggering for off-camera flash). This would eliminate the need for two receivers and the associated batteries, but would leave you still controlling the flashes optically. Keep in mind that as you start to use modifiers, such as a softbox that may require you to bury the flash inside it, the line of sight limitations of optical control will probably convince you to eventually move to radio anyway.
I have two Yongnuo Speedlite YN-568EX-II units. I can use them as radio/receivers between the two, ...
Actually, you can't. The YN-568EXII has no built-in radio capability. It can do "smart" optical slave/master, and that may be what you're thinking of, but it's a light-based system and has weaknesses of range and line-of-sight requirements. But the system will let you use TTL and HSS.
... I'm currently thinking of the Yongnuo YN-622N-TX - but is this compatible?
Not directly. You would need to add YN-622 transceivers to the feet of your YN-568EXIIs to use them with a YN-622-TX. But, basically, any TTL/HSS radio triggers made for Canon (assuming you are shooting Canon) should work with the YN-568EXII, although sometimes 3rd-party gear may not work with other 3rd party gear, the way it works with Canon gear. So sticking with Yongnuo may be your best bet. But it's not your only one.
You just have to make sure there's a dedicated transmitter/controller for the camera hotshoe, and receivers for the flash feet, and that both have all five of the contacts on the hotshoe/pins on the feet that Canon uses for camera/flash communication.
You could probably also use (among others):
Radio trigger technology moves swiftly, and new triggers are coming out all the time. Two of the best sites to keep up to date on what's coming out would be the Flash Havoc and Lighting Rumors blogs.
Today, most folks prefer speedlights with built-in radio triggers as being more convenient than using add-on triggers. You don't have to remember to bring along triggers and batteries for them; and if you're getting super-cheap manual-only speedlights, they often offer remote power control and possibly HSS, over manual-only speedlights without built-in triggers. The drawback, though, is that you will be tied into whatever triggering system is built in.
