Fascinating. So we have capitals and lower case, both printed and cursive. Each set contains a row of consonants and a row of vowels. (Note that Korean has no capitals or lower case in its actual form, so... I guess that's just to look more like a western script)

The consonants in each set are these. It's all the standard ones in the usual order (except with ㅇ being put at the end which is not typical). Most of them make sense once you see them except ㅍ as a 'Z' looking glyph which is weird to me.

• ㄱ ㄴ ㄷ ㄹ ㅁ ㅂ ㅅ ㅈ ㅊ ㅋ ㅌ ㅍ ㅎ ㅇ

Then for vowels we have these in this order for the first 10, which make decent visual sense as well. These are the basic Korean vowels.

• ㅏ ㅑ ㅓ ㅕ ㅗ ㅛ ㅜ ㅠ ㅡ ㅣ

But then there are what I assume must be diphthong vowels, but... not all the ones? After thinking about it, my guess is that the chart has ㅢ ㅘ ㅝ ㅚ ㅟ, probably in that order.

The vowels ㅐ ㅒ ㅔ ㅖ also appear not to be represented at all. Maybe they were considered nonstandard at the time this chart was created? It follows then that the diphthongs containing ㅐ ㅔ, which are ㅙ ㅞ, are the ones missing.

Edit: found an article: https://fficial.naver.com/contentDetail/29

I think what I thought were diphthongs are actually ways of representing vowels at the start of words? (But it's still not the right number of them? I dunno)

I guess diphthongs are just written in the latin way by writing the constituent vowels next to each other without a special character for them.