Parts you may need to rebuild a diff:

4348679 pinion gear (spider, 2 reqd) $22.50 ea
4659502 gear diff (axle side gear, 2 reqd) $36.70 ea
4659142 Shaft pin (spider gear shaft 1 reqd) $10.70 ea
4412521, spider gear washer (2 reqd) $1.60 each
4504568 washer shim kit for diff (2 reqd) $9.90 ea
4800059-AA Bracket, pin retainer. (2 reqd) $4.20 ea
 
 
 

To remove the differential, remove the stamped steel cover (10 mm socket) then the 4 long bolts on the pass side bearing support, and the 6 shorter bolts from the drivers side (shown above still installed) All three of  these items are installed with only silicone. Use normal practice to remove the cover. It will spill fluid. To remove the axle bearing supports, the must be twisted back and forth to work them out. To move the Speedometer side (shown loose above) I stuck a wooden hammer handle into the speedometer gear hole and used it to lever the support back and forth, breaking the silicone seal.  On the drivers side, I put two of the bolts back in the holes, but made sure they did not go in far enough to keep the support from turning when I used a small pry bar to apply a twist to the housing. In the picture, I am pressing down on the bar, twisting the support.  The diff. will fall out when both are loose.

Note. It is not required that you remove the differential but you MUST remove the steel cover and inspect the unit for damage. The spider gear shaft will fall out:

Pin is slipping out!

This one is about to damage the case! It looked like this when I pulled the cover!
 

Above picture was sent to me from Young Friend who really like to spin the tires. The pin came out here, can you imagine what happened inside! I have seen fluid leaking out of the bell housing where the pin came through case on the converter side!

Finally Chrysler has come up with a "Fix" for this problem, a bracket to keep the pin in:
Part number is 04800059AA, cost about $3.00 each (You need two)

pin retainer brackets: 04800059AA
spider gears: 04348679AB
Spider gear shaft: 04659142
shim washer kit 04504568

With these simple brackets, I leave out the little "tapped in" retainer pin completely, this allows the shaft to rotate in the case some, basically allowing the spider gears to wear on the shaft all the way around rather than in one spot (Kind of like valve rotators)
See, with the shaft rigidly mounted to the case, it always sees the force in exactly the same spot, causing wear in one spot, and it will bend. Also, the shaft is chrome plated, and once the chrome is worn through the shaft is going to chew itself up. With the abilty to rotate in the case, the gears bear on a differnt side of the shaft every time you punch it, I feel this greatly extends the life of the chrome plate and the shaft. This shaft is one year old:

You can see that it had sheared the little retainer pin anyway. This pin had a "spiral grove" cut in it to promote lubrication of the gears. It did not help :-)

A link to a "limited slip" unit for this differential

A link to a Phantom Grip install on the 413 Auto

A Link to the Phantom Grip install in a 555/568 5 speed

A picture of the Future Auto Limited slip unit apart:

That is a oil bottle cap next to it. A clutch disc is on the back side of the axle gears

Here is a link to the Phantom Grip install

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