Is it possible to use the Unicode superscript and subscript parentheses as 
superscript and subscript operators? That is, ⁽…⁾ having the same function as 
^{…}, and ₍…₎ as _{…}.

It might be something like the following, but somehow bundling the last part of 
\letcharcode into a command.
\appendtoks
 \catcode`⁽=\activecatcode
 \letcharcode `⁽ ^{
 \catcode`⁾=\activecatcode
 \letcharcode `⁾ }
 \catcode`₍=\activecatcode
 \letcharcode `₍ _{
 \catcode`₎=\activecatcode
 \letcharcode `₎ }
\to \everymathematics


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