The question is rather simple. Why 'contre' is used in the following sentence?
Votre courrier a été distribué à son destinataire contre sa signature (suivi d'une lettre recommandée).
[I understand the desired meaning.]
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Yes, this seems rather counter (contre?) intuitive, doesn't it? :)
The semantic link you're missing is that of reciprocity. An action earns a reaction. A delivery earns a confirmation.
In the WordReference entry, the closest definition to the one used here is the fourth: "for, in exchange for":
Je donnerais dix ans de ma vie contre une bonne santé.
The idea is that of an exchange; "contre", in this usage has the following meaning.
(TLFi) III.− Marque l'échange, le rapport de deux grandeurs.
Échanger un sac de billes contre un couteau; parier à dix contre un
In English this meaning could be rendered by "in return for". The word does not make explicit that the signature is absolutely necessary.