I am looking for a single word that describes the act of aligning a
stack of papers (and "align" doesn't work).  You know, when you have a
stack of papers that aren't aligned properly, you raise them
vertically (perpendicular to the surface of the table), hold them by
the sides, and drop them onto the table.  After repeating the process
a few times, the papers are aligned, nice and neat, one behind the
other.  I am looking for a word - a single word - that describes this
act.  I want to be able to say, "Hey Doug, _____ those papers," and
have Doug know exactly what I want him to do.

Align, stack, shuffle, organize - none of them work.

I've thought long and hard about it, and I don't think this mystery
word exists...  See if you can convince me otherwise!  :)

I don't know of a one-word term for this, but when I align a stack of
papers I say I am "squaring them up."

when you're at a complete loss for 'le bon mot', MAKE one up...it
worked for Lewis Carroll. In the spirit of the festive season I offer
you :

Shoggle 

\Shog"gle\, v. t. (See jog, joggle, shuffle ) [Obs. or Prov. Eng. & Scot.]

1. To manually create an orderly stack of previously wayward papers.
2. The act of shaking a Starbucks sugar bag back and forth before
tearing the corner off, usually done between thumb and forefinger.

Phink

I posited this at dinner and the general consensus was half came up
with Pinkfreud's response and the other half went with "trueing up the
edges." I did not offer Pink's expression for voting, they came up
with it themselves.

I like "square up" myself. And shoggle is a delight!

One of the definitions of "collate" seems to fit this process.

How about: "Hey Doug, straighten those papers." 

Rainbow~

It's known as jogging in the trade.  In fact, you can buy an electric paper jogger:

http://www.officezone.com/jog1.htm

Ditto. In the printing business, whether by hand or by machine, the
process is known as "jogging.' Tell Doug to jog the sheets :-)

JOGGING!  Wow, I never would have guessed it!  Thanks for the info -
it's nice to finally know the answer!

LOL @ shoggle  :)

Yep - from the Merriam Webster site:

4 : to align the edges of (piled sheets of paper) by hitting or
shaking against a flat surface