In fact you just say "having reversed (row-wise) homology" in each node of the usual snake lemma, and you have a generalization of it that covers all the above mentioned cases.

In the SRS to LES it should be LRS as the final sequence could have reversed homology in some of the terms.

How do such structures arise? When they arise, what questions does one want to answer? Knowing this might be useful in providing direction and focus.

@JohnPalmieri essentially anywhere you rely on exactness (as a containment) in a proof but not on $d^2 = 0$ equation itself, then the theorem can be generalized in some way to include sequences with reverse homology.

@DanielDonnelly That's not John's question. He's asking for examples of this phenomenon in the first place - that is, why should we care about this additional generality?

@DanielDonnelly The snake lemma goes by "lemma" for a reason. It's a tool. Why spend any time and energy generalizing to something without apparent use? There is no glory in this: The snake lemma has a name not because it is a great achievement, but because it is so useful. The answer to your question then suggests itself.

@BenSteffan a tool that can do much more now.

@DanielDonnelly ...says you, who has yet to provide us with even a single salient application. If you can't answer John's and Noah's questions, then, at the risk of repeating myself, the answer to your question suggests itself.

The theory generalizes in more places than one and the Snake Lemma is used everywhere like with Yoneda's Lemma in category theory. Let its effect on generalizing the entire theory speak for itself.

@DanielDonnelly "The theory generalizes in more places than one" Cool, so you should be able to give at least one interesting example. "Let its effect on generalizing the entire theory speak for itself." This sort of grand sweeping claim absent any justification or example misses a key part of why the "abstract" approach in mathematics is so successful: we don't just pointlessly generalize existing definitions purely for the sake of generality itself. (A certain remark by Girard occurs ...)

I think that "Why hasn't [X] shown up naturally" is not a good question. I can't imagine what answers it might have beyond "because it hasn't shown up naturally."

@JohnPalmieri added Down2Earth example at the bottom.

@JohnPalmieri you take each HA proposition, and see where you can get away with reverse homology instead of full-fledged exactness. You can then operate on reverse and forward sequences as units, but you can't mix forward and reverse in the same sequence and retain an abelian category because the kernel/image containment direction can be reversed randomly, so is not fixed to one and so there is no consistent direction. I'm not sure if there could be a morphism between forward and reverse complexes ("sequences")