What is an "antonym" of "rummage?" I cannot seem to figure out a word which means "searching systematically and tidily through a mass or receptacle."
That would be...
Sift every grain of information until you find the answer.
He spent hours sifting through all the documents relating to the case.
They sifted through the rubble, searching for signs of life.
I'm glad you put the scare quotes round 'antonym'.
The only single word I can think of for 'search [X] thoroughly' is 'comb'. Which has various senses that you might not want. And would only work in certain situations (ie only have certain collocates).
If an expression will do, 'go through with a fine-tooth comb' is spot on.
There's always something like inspected, as in "He inspected the contents of the desk thoroughly."
Enumerate will work in some contents. And as bib mentions, catalog can work as well.
Further possibilities include: pore over, peruse, scan .
It is also worth noting that rummage can also be used to mean in a sense close to what you are looking for (in addition to the unsystematic version you describe):
From the OED :
(rummage, v.) trans.
To scrutinize, examine minutely, investigate.
1864 J. H. Burton Scot Abroad I. v. 305, I am inclined to concur in Pasquier's silence, having rummaged his ‘Recherches de la France’.
1907 Life Sept. 71 The shallow mind rummages the fields of fiction with a morbid desire to see how this or that ends.
1975 Gen. Syst.20 108/1 The search for causes must first rummage the material factors, not the social-organizational or ideological factors.
2004 W. M. Billings Sir William Berkeley p. xi, I rummaged the records in search of a deeper understanding of seventeenth-century Virginians.
I would suggest ransack, in the non-plundering sense:-
To search or examine thoroughly.
To search through every part of (a house, box, etc); examine thoroughly
To search thoroughly or vigorously through (a house, receptacle, etc.).