The rects themselves have nothing to do with TSC direction; direction is something that gets checked in the code to determine what rects to pull. Rects are somewhat difficult to visualise, but think of them as defining four infinitely extending lines; the line display_L draws a vertical line (for the left side) and everything to the left of that line is disregarded. display_U draws a horizontal line, and everything above is disregarded. Continue this logic with the other two and what you're left with is a rectangular area.
As for what you've mentioned, after a bit of searching I found it in one of rune's old tutorials. The proper order is, in fact, LURD, and I'm not sure why he put them in LRUD order (possibly to group them by x/y, I dunno). Everywhere else in the compendium where it lists the actual offsets of these rects they're properly ordered as far as I can see.
In fact, if you wanted to be weird, you could rewrite an NPC's rendering code to read the rects in a different order, how it works is there's four different "containers" for the rect data, let's say, and the rect data to access is all in a row, which saves space when trying to read it. LURD is the convention in cave story though, so that's all you really need to worry about.
Finally, when you're dealing with rects and such in the .exe, remember that all the numbers are hex; hence, 16 = 10, 32 = 20 and so forth.