How to create an index entry for "|| operator" in LaTeX?

Question

Following the \index quoting rules (prepend a reserved character with a double-quote), I do

\index{"|"| operator}

but the entry never shows up in the index. Doing these (similar) things does work as expected:

\index{"!"! operator}
\index{"| operator}

So what is special about ||? Needless to say, it also doesn't work without double-quotes.

It seems that this only happens when the hyperref package is used.