Reinstalling blank keycaps

Probably a stupid question, but lately many switches on my Model01 send duplicate keypresses and keycaps are getting dirty in general. I would like to take the keycaps off, put them in a bag with dental tablet solution for cleaning and vacuum the keyboard itself, however I cannot figure out a way to reinstall them back after I do it.
Normally, keycaps have row numbers printed on the backside, such that install is easy, however it’s not the case with blank Model01 keycaps and it seems that each keycap, except for some exceptions, has a unique spot on the board. I don’t think I will ever figure out which keycap would go where by my own once I mix them all in a bag.
In fact, even when removing a single keycap for cleaning I often mess up the rotation of it.
The only strategy I can come up with is to put the qwerty set next to me and try to match each keycap to the qwerty one to determine the position, but that seems to be overly complicated and slow. Is there an easier solution?
TL;DR: How to figure out which blank keycap goes where?

The keycaps are numbered on the bottom. The details are here.