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The primary goal of this paper is to study Spanier–Whitehead duality in the K(n)-local category. One of the key players in the K(n)-local category is the Lubin–Tate spectrum 𝐸𝑛, whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that 𝐸𝑛 is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group 𝔾𝑛 of the formal group in question. In this paper we find that the 𝔾𝑛-equivariant dual of 𝐸𝑛 is in fact 𝐸𝑛 twisted by a sphere with a non-trivial (when 𝑛>1) action by 𝔾𝑛. This sphere is a dualizing module for the group 𝔾𝑛, and we construct and study such an object 𝐼𝒢 for any compact p-adic analytic group 𝒢. If we restrict the action of 𝒢 on 𝐼𝒢 to certain type of small subgroups, we identify 𝐼𝒢 with a specific representation sphere coming from the Lie algebra of 𝒢. This is done by a classification of p-complete sphere spectra with an action by an elementary abelian p-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the K(n)-local Spanier–Whitehead duals of 𝐸ℎ𝐻𝑛 for select choices of p and n and finite subgroups H of 𝔾𝑛.