Headers on a table with two double column of letters with arrows between

Question

I want two double columns of letters with arrows between. The headers should be centred above each of the double columns:

Header 1        Header 2
  A  B     <=>    C  D
  E  F     <=>    G  H

I'm using

cc@{$\Leftrightarrow$}cc

for the tabular body template and \multicolumn for the headers. However, when I write \multicolumn{2} the header is centred over the first double column plus the arrow.

    Header 1       Header 2
  A  B     <=>    C  D
  E  F     <=>    G  H

Is there any way to achieve what I want without manually inserting arrows in every row?

\documentclass[11pt]{article}

\begin{document}

\begin{tabular}{cc@{$\Leftrightarrow$}cc}

%Header centred over first double column of letters, then omit arrow, then header centred over second double column of letters.
\multicolumn{2}{c}{Header 1}
    &\multicolumn{2}{c}{Header 2}\\

%double column of letters, then arrow, then double column of letters
A& B& C& D\\
E& F& G& H\\
\end{tabular}
\end{document}

Answer 1

Part of your problem comes from the headers being wider than the sum of the widths of the two columns underneath.

Here is a simple solution with a supplementary column between the two groups:

A simple solution with a supplementary column between the two groups:

\documentclass[11pt]{article}
\usepackage{array}

\begin{document}


\begin{tabular}{cc >{$\Leftrightarrow$}ccc}
%Header centred over first double column of letters, then omit arrow, then header centred over second double column of letters.
\multicolumn{2}{c}{\makebox[0pt]{Header 1}} & \multicolumn{1}{c}{}
    &\multicolumn{2}{c}{\makebox[0pt]{Header 2}}\\
%double column of letters, then arrow, then double column of letters
A& B& & C& D\\
E& F& & G& H
\end{tabular}

\end{document}