Indexing an argument to a new command produces odd results

Question

Consider this basic .tex file:

\documentclass[12pt, a4paper, draft]{book}
\usepackage{makeidx}
\makeindex
\newcommand{\withIndex}[1]{#1\index{$#1$}}
\newcommand{\myIndex}[1]{\index{$#1$}}
\begin {document}
\section{Test section}

Some test text \index{text}.

A $\Downarrow$ \index{$\Downarrow$} represents....\\
Another \withIndex{$\Downarrow$} represents...\\
Perhaps $\Downarrow$ \myIndex{$\Uparrow$} represents...\
\printindex
\end {document}

I have the \withIndex command defined on line 4 in my document, and it works well for the majority of symbols. However, when certain symbols are used with this command odd results are produced. Calling \withIndex($\Downarrow$} in the document body puts the key ${\Gamma}B37F$ into the index (to give another example, indexing $\langle$ in this way gives the key 'hA').

Naturally, I want to see the \Downarrow symbol appear in the index (or the left angled bracket in my other example). The \myIndex command produces the same result, but \index{$\Downarrow$} gives the correct output. Why do I get such strange index output for these particular symbols with the \withIndex command?

This question is very similar to Symbol index sorted by occurrence. Look if it helps you. — egreg, May 13 '13 at 23:35
You shouldn't add $ in the definition of \myIndex and \withIndex, though, if you plan to call \withIndex{$\Downarrow$}, because this would result in $$ in the index file. — egreg, May 13 '13 at 23:47
Ah yes indeed, thanks for pointing that out. I'll make that edit in my file. — John Pirie, May 13 '13 at 23:49

Indexing an argument to a new command produces odd results

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