What's the right spelling of synchronized (like in synchronized swimming), and are there any differences between the British and American English?
Syntax will come into play when you present a complete sentence. A single word cannot have a "correct" syntax. Differences between British and American English will manifest in the spelling. US will use "synchronized swimming", British English will use "synchronised swimming".
With regards to syntax, you may need to provide a bit more context to warrant a valid answer.
British spelling is "synchronised" and American is "synchronized" so the correct spelling will depend on yourself and your intended audience.
Both are in common use in UK English.
Oxford University Press style is -ize in almost all cases: http://blog.oxforddictionaries.com/2011/03/ize-or-ise/
But their dictionaries accept either spelling.