The symmetry between introduction and elimination rules for the logical constants is called harmony. The idea of logical harmony has been defended by several logicians, including Gentzen and Prawitz, as being a requirement for a proof-theoretic justification of logic. Harmony guarantees that the introduction of a logical constant is conservative with respect to implication. Michael Dummett has taken the argument further and claimed that any language, including natural languages like English, should have harmonious and stable rules for its terms. Dummett proceeds on this basis to argue that since classical negation is not harmonious, it has no defensible meaning, and he takes this to be an argument for adopting intuitionism and a verification based semantics for language.
These claims are disputed. Ian Rumfitt argues that harmony is overkill as a condition of admissability, and that defective logical constants such as Arthur Prior's 'tonk' can be ruled out because they lack truth conditions.
There is quite a good explanation of this issue in Nils Kurbis "Proof-Theoretic Semantics, a Problem with Negation and Prospects for Modality" Journal of Philosophical Logic 44 (6):713-727 (2015) which can be found on PhilPapers.org at https://philpapers.org/rec/KRBPSA
Other useful references are:
Steinberger, F. (2011) “What harmony could and could not be”. Australasian Journal of Philosophy 89: 617-639; and
Rumfitt, Ian (2016) “Against Harmony”. Forthcoming in Robert Hale, Crispin Wright, and Alexander Miller, eds., The Blackwell Companion to the Philosophy of Language, 2nd edition. Oxford: Blackwell.