Interesting idea. Presumably the games would alternate white/black/white/black...
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Personally I never ask for a rematch and almost never accept one. IMHO any claims of rematch aficionados about etiquette, morals and fairness are absolutely ridiculous.
However stretching my compassion to its maximum here is an idea - chess.com could implement an option to create open challenge for a match of N games where N is a number between 2 and 10 for example. May be it will be of some interest in addition to single games and tournaments.