I've also seen, with many piyutim/selichos, something like this:
Chazzan: line 1
Kahal: line 1, then line 2
Chazzan: line 2
Kahal: line 3
Chazzan: line 3
etc.
I suspect that with a lot of these piyutim, the original idea was that the kahal should just reply with the repeated phrase of each line (e.g., אמרו לאלקים or מראה כהן), especially back when siddurim or machzorim were rare and therefore you couldn't expect everyone to know all of the words. Later, when it became more common for the kahal to say all of the words, that mutated into something like the above, because they'd want to be able to also listen for the chazzan's recitation/singing of the words (rather than just saying them with him).