Bingo bango bongo, I never noticed that. I tried saying 'fee fi fo fum' backwards and it took me about 10 times to get it right. There's almost a musical cadence to the usual order that just sounds off key when said backwards.
This is language-dependent. The Japanese syllabary is sorted in the order a-i-u-e-o, which is weird to the Western ear, but as ubiquitous as A-B-C in Japan.
It's probably not unwritten. It's similar to how we conjugate irregular verbs like "sing/ring" or "beget". The topic in linguistics is called vowel gradation [0].
In a series of words which differ only (or mostly) by the vowel used, the order should be e, i, a, o
- tic tac toe
- flim flam
- ding dong
- king kong
If you doubt it, try saying the opposite and hear how odd it sounds: The clock went 'Tock Tick'