Why not leave it in the RIGHT side as that's when it locks up?
Without being there to see it, all I can do is try to explain how trans works and then you need to inspect carefully.
As I mentioned before, there are 3 sliding gears that the shift forks move.. They have "dogs" or protrusions on them that lock into holes or other "dogs" on the adjacent gear.
In the drawing below, the sliding gears are marked up in blue with the dogs as blue dots. The yellow squares are the adjacent holes/dogs that the sliding gear dogs lock into which puts it in gear.
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When in neutral, as shown in the drawing, NO dogs are engaging with an adjacent gear. This is why trans spins freely in neutral.
When you select 1st gear, the groove in the shift drum moves fork #3 to the right. This engages the dogs (blue dots) on the sliding gear with the holes (yellow squares) in the free-spinning 1st gear, locking it to the shaft. All the other sliding gears stay centered in their "neutral" position.
When you select 2nd gear, the #3 shift drum groove moves fork #3 back to the left, disengaging first gear. At the same time, the #1 shift drum groove moves its sliding gear to the left, locking in 2nd gear. Note that only ONE set of dogs can be engaged at a time or the trans will lock up. That's why the shift drum groove #3 disengages 1st gear at the same time shift drum groove #1 engages second gear.
When you select 3rd gear, the shift drum groove #1 slides the sliding gear to the right disengaging 2nd gear and engaging 3rd gear. (Shift drum grooves #2 and #3 keep those gears in the neutral position)
When you select 4th gear, shift drum groove #1 moves that sliding gear back to the left into its neutral position, disengaging 3rd, while simultaneously shift drum groove #2 slides its gear to the right, locking in the 4th gear dogs.
Finally the shift to 5th simply has groove #2 moving the sliding gear to the left which disengages 4th and engages 5th. Grooves #1 and #3 keep their sliding gears centered in their neutral positions.
Summary is that in neutral, all the sliding gears are held by the forks so that no dogs are engaged.
As you shift through the gears, the shift drum rotates and the grooves in the drum slide the sliding gears side to side, sequentially engaging and disengaging the dogs for each gear.
If the shift drum isn't centered properly, or the forks get bent or mixed up, or the gears on the shafts aren't assembled correctly you can have a situation without neutral, or worse, two sets of dogs will engage at the same time locking the trans up.
Hopefully this will help you see what is going on and why the trans locks in the right case. If the forks are in the right places and not damaged (usually pretty obvious if a fork got bent), then my guess is something is allowing the shift drum to move off-center or one or both of the shafts are shifted off center in the cases.
The thing that centers the drum is the "horseshoe" clip and shim on the end of the drum under the plastic cap outside the left case.
The clutch shaft is pulled to the right by the inner basket nut. The ball bearing in the right case keeps the clutch shaft offset where it belongs.
Similarly, the sprocket shaft is held in place to the left by the sprocket nut. The shaft is located by that ball bearing.
If either of those ball bearings are not the correct parts or not installed properly, one of the shafts could be shifted sideways which can engage dogs when they are not supposed to be engaged.
Hope you figure it out soon!
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