Some 釜 come with their 鐶, but 鐶 can be 道具 in their own right. I like the comparison to a 茶入 and her 仕覆.
You'll find them made either from cast iron (by 大西清右衛門 or alloys (by 中川浄益)
Types
The 水屋鐶 are made of brass which is more likely to bend. Because of this, they are twisted by a full 180° (as opposed to the about 90° for conventional 鐶).
蜻蛉: Since the dragonfly is a 勝虫 that always flies forward, so do these fellas!
They live in the 炭斗 resting against the charcoal, like in 後炭 even for 初炭 (I presume that otherwise they'd hang down and that's no good).
Since they have a front and a bag, you need to deal with that:
Both lie in the 炭斗 with the front facing towards you.
When you take them out, you need to turn the one that goes to the left of the 釜.
Normally, when putting them down, you'd turn that one back again, and put them on the 畳 facing upwards.
However, this is my idea and KM26 approves, it's nicer if they don't come to a rest. So instead of laying them on the 畳, stack them against each other, facing you (like a 蓋置).
When putting them back in the 炭斗, turn the left one again. My conjecture is that now they be face down; or better still: after 後炭 only they land in face down.
You use them a bit like the big 鐶 with 釣り釜 in that you can't rest them against the 釜. This changes the 手前 a little bit:
Before attaching them, slide forward, sort 釜敷, then attach, lift and shift while holding them.
When removing, you can't adjust the 釜 position. Put the 釜 where it belongs, detach. Then 釜敷, then shift backwards (this is a conjecture of KM26 and I; but the other way gives a lot of emphasis to the shifting).