Why do you write down a write-up and not vice versa?

I can understand why you write something 'down', because as you write, the text goes down the page. But why then do we have a write-up?

• "Write down" combines the Old English verb writan (to score, draw, or write) with the Old English adverb dūne (from the hill/downwards).

Historically, this directional "down" emerged to signify fixing abstract speech into a physical, concrete state on a page. Hard copy. This is part of a more general usage, making something (literal or abstract) secure, fixed in place, permanent, enduring:

down [on paper]:

put down what he says Merriam-Webster

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down: firmly in place or into position:

I put the loose floorboard back and nailed it down. [Cambridge Dictionary]

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write it down before I forget it Ray Carney, Boston University: The Path of the Artist, Part 2 {though this hardly needs a quote; there are many other examples on the internet}]

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I grabbed a napkin and jotted it down. [Times, Sunday Times (2007); via Collins Dictionary]

So the 'down' part retains the broadened sesne of fixed securely.

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• The Online Etymological Dictionary gives the etymology of the compound noun write-up:

write-up (n.) "written account or description," by 1882, from the verbal phrase; see write (v.) + up (adv.). To write up is attested by early 15c. as "put (a full account) in writing." It is by c. 1500 as "enter (someone's) name in a list or roll" for a particular purpose.

Notice the 'full account' mentioned for the transitive multi-word verb 'write up'. Here, the broadened sense of 'up' speaks of thoroughness; it is usually called completive up, and occurs commonly in transitive multi-word verbs (tidy up, clear up, clean up, eat up ...) and the [often] idiom shut up shop. See John Lawler's answer here.

Detailed explanations of why these common adverbs took on these specialised roles would need a vast amount of research, and theories would probably be contentious. See jsw29's comment under John Lawler's answer. But 'getting something down on paper' does have a reasonably convincing directional tie-in.