Phonetic Japanese does actually have an order, although it was never taught to me in textbooks.
It is a ka sa ta na ha ma ya ra wan, i ki shi chi ni hi mi ri, u, ... , e, ..., o, ...
If you say the whole thing, it sounds interesting, like a meaningless poem.
- akasatana hamayarawan
- ikishichi ni himiri
- ukusutsunu fumuyuru
- ekesetene hemere
- okosotsuno homoyoro
No matter what script things are listed in (kanji, katakana, or hiragana) it is ordered by this.