Learn Finnish: Hinta on korkea tai matala päivästä riippuen. - The price is high or low depending on the day.

Both words come from the verb riippua (to depend / to hang) and behave grammatically the same way. While riippumatta means "regardless of" (literally "without depending"), riippuen means "depending on". Both act as postpositions that force the preceding noun into the elative case (-sta/-stä), which is why we have päivästä here.

What grammatical form is 'riippuen'?

It is the second infinitive (often called the E-infinitive) in the instructive case. It is formed by taking the strong stem of the verb riippua and adding -en. This structure is used to express the manner in which an action happens, functioning very much like the English "-ing" participle here.

Why are the adjectives 'korkea' and 'matala' in the nominative case instead of the partitive?

Because hinta (price) is treated as a distinct, countable concept in this context rather than an undefined mass. When a subject is a specific, indivisible whole in the nominative case, the adjectives describing it (the subject complement) must also take the nominative case. You would generally use the partitive if the subject was an uncountable mass noun (like water or time).

Can I swap the word order and say 'riippuen päivästä'?

No, you should stick to päivästä riippuen. Even though they derive from verb forms, both riippuen and riippumatta function as postpositions in these phrases, meaning they naturally belong directly after the elative word they modify.