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    $\begingroup$ This is not very helpful, but I think it's more natural in this case to do Cases[Flatten[Apply[Outer[Intersection, ##, 1] &, input], 2], {__}] at the end. $\endgroup$ Commented Mar 23, 2016 at 21:50
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    $\begingroup$ As possibly another unhelpful comment: is it possible to generate the full equivalence relation at the beginning, before you've generated the individual ones? It seems like it would be better to combine the conditions used to generate the equivalence classes first before generating the lists themselves. That way you can get output without generating input first. (This is still good question and should be answered. I figured I'd toss this out there anyway.) $\endgroup$ Commented Mar 23, 2016 at 21:53
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    $\begingroup$ Actually, Cases[Apply[Outer[Intersection, ##, 1] &, input], {__}, {3}]. $\endgroup$ Commented Mar 23, 2016 at 21:56