How to reassemble a Shimano Alivio Shifter
It just so happens I had one of these land in my garage for a fixup today.  So I've taken some photos of how it went.

tl;dr Answer Root cause lubrication. The pawl for up-shifting wasn't engaging the teeth.  

This is for a Right hand shifter, for the REAR mech.  I assume a Left shifter for the FRONT mech is similar inside, with fewer teeth.


Remove the inner wire - it gets in the way during disassembly.  If its at all rusty or hard to pull through, then dispose and fit a fresh one.  They're cheap.  Leave all the outer housings in place unless they need replacing too.
Remove brake/shifter from handlebars - this probably means removing the grip and anything else in the way.  
Clear your workspace - don't want to loose any parts.
Remove FOUR screws - two screws from underneath, one shouldered screw from on-top by the brake adjuster, and the brake reach screw.  Also remove the inner cable adjuster by unthreadding it, but don't lose the loose spring. Put aside.  
Using a spudger or a flat screwdriver, separate the top cover near the brake wire adjuster.  It should rotate up around the 7 segment gear display and look like this one apart.
  Note this shows an inner wire in place - I've used a short piece as a demonstrator.  The clear plastic 7 slot display is on the main body in this photo.  It should slide out easily, and fit it to the cover instead.
Next lift off the small white nylon lever from the center of the main shaft.  Put aside.

Remove the last philips screw underneath.  Put aside.
With a suitable sized spanner, remove the big black hex bolt underneath.  You'll need to keep your finger on the other end of the main shaft, because the nut will rise out of its recess and simply spin.  Then remove bolt and nut, put aside.  You can take off the bottom cover too.

The brake lever and other bits of plastic should just come off now.  Put them all aside.  We want the main guts of the shifter, which look like this:


Now you have to find out why its not working.  For me, I could shift to a higher gear, but not to a lower gear.  The cause was this pawl around the back, that was not dropping down into the teeth and pushing.


                
            
So I fixed mine by washing with a small paintbrush and turpentine, allowing it to dry, and then wiped in a small amount of Lithium Grease using a finger.  I could have sprayed it but then these photos wouldn't work too well.

Reassemble is the reverse process.  Start by refitting the thin plastic ring, and then work down towards the bottom.  Insert main bolt.
Then flip the whole thing, and refit the bottom cover then the brake lever.  Make sure to get them lined up like this and that there are two holes the inner cable will pass through to get out - they have to line up.  Also, in other angles the upshift/finger lever will obstruct the brake or the bars.

Tighten up the main bolt (not too tight!)
Refit the white plastic pointer thingy.  This is best done in 7th gear like this:
  
With the display window fitted to the top cover, slide the display window into the bottom cover. Once its home, you should be able to click down the rear of the top cover.
Reinstall 4 screws and the cable adjuster.  Fit an inner cable and test.   Note if you don't have an inner cable, testing might get it stuck!
Refit to bike, recable, and all the usual things.  




By Murphy's Law I had to do the left hand shifter too the next day.  Process was identical but mirrored left for right.  There are only three positions on the left shifter, so far fewer teeth on the internal cogs.   

No pics of the left shifter because someone else had used adhesive glues to stick several broken pieces back together.

The underlying problem was identical - a pawl was stuck in a position where it could not contact a tooth.  Turps and lube was the fix.

One thing that messes me up is trying to count the effective positions, in that you don't count Clicks or Presses of lever.  Instead, the first position is #1 and one click takes you to position 2 etc.  So a three position left shifter has TWO clicks.  I spent an embarassingly long time trying to find the third click.