Can one develop a theory of étale homology via étale cosheaves? The hope is that this would, for example, return the Tate module (and not its dual) for an elliptic curve, and it would return group homology for $\rm Spec$ of a field.
The first step in this direction is to notice that you can define an étale cosheaf on $\rm Spec$ of a field, in which the sections are coinvariants, rather than invariants, of the associated Galois module. In this case I believe we would recover group (Galois) homology.
(See Cosheaf homology and a theorem of Beilinson (in a paper on Mixed Tate Motives) if you don't know what cosheaves are.)
I'm not expecting this to prove anything new, but it would make certain formulations nicer (for some definition of "nice"). E.g., I'm hoping we would get theorems that look similar to the relations between singular homology and cohomology, and we would get a comparison with singular homology for finite coefficients.
I feel like this should work, though I'm not entirely sure about the ability to cosheafify.