How much is cohomotopy dual to homotopy?

To what degree can we dualize theorems regarding homotopy into theorems about cohomotopy (or is there a good source that tries to do this)?

For instance, is there some kind of Hurewicz theorem relating cohomotopy and ordinary cohomology? Is there a “cohomotopy extension property” (something that applies when relative cohomotopy groups are trivial)? If two spaces are cohomologically equivalent and have some property in cohomotopy analogous to simply-connected, are they cohomotopy equivalent?

Thanks, this is primarily a reference request, however there is the possibility that all this is impossible so no such reference exists, which would also be an acceptable answer.

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The homotopy groups can be written as covariant homotopy invariant functors $\pi_n:\mathrm{Top}_\ast\to\mathrm{Set}$. If we were to consider contravariant homotopy invariant functors $\pi^n:\mathrm{Top}_\ast^{op}\to\mathrm{Set}$, we would obtain the cohomotopy sets. How dual is it? Well, $\pi^n(S^m)=\pi_m(S^n)$. If $X$ is a CW-complex of dimension (at most) $n$, then $\pi^p(X)\to H^p(X)$ is a bijection. See the nlab. As for whether or not a “cohomotopy extension property” exists, I don’t know; it seems like an interesting thing!