This is probably irresponsible but I decided to place a link to a raw pdf file entitled "Self-dual formulation of homology, exact couples and spectral sequences". It is a beamer presentation and it is awful: it does not contain a single word, just a pretentious panoply of diagrams.
The reason I decided to do this: I think the primary drawback of spectral sequences is the nightmare of indices, which is moreover not justified at all. The essential content of spectral sequences does not have anything to do with gradings a priori. So I decided to work out the construction of exact couples in the non-graded setup.
The result was spectacular in one respect: it shows that even if you get rid of gradings, the thing is still quite complicated :D
Still, I find it amusing to see what remains when you throw all the grading stuff out.