I finally tore into this.
Turns out the reverser is the entire center housing between the motor and transmission.
It is between the clutch and transmission but it's deceiving because the shifters don't go into the transmission. The transmission is actually in with the rear end.
To work on the reverser you need to take off the cover with the shifters, and then remove the rear end/transmission, and then remove the housing with the reverser from the motor. That whole thing needs to come out.
I took off the shifters and rear end today. And took out the reverser, except it's just one end of it. You cant get to the other end without removing the housing from the motor. They put one set in the front half of the housing and the other set in the rear of the housing so you really need to pull the whole thing. I don't see any bent pipes or anything but it don't matter at this point to put it together without checking the front wouldn't make any sense at all.
Once you get the back off there's not that much more work to removing the whole housing. This design is actually good in a way. When I put it back, I can put the entire reverser together, control block, tubes, and all, and then put the whole unit back in the tractor. Having taken it apart I found that there is a plate behind the control block that comes off (must be removed to get the reverser out of the housing), but removing it if your not taking the reverser out would be a bad idea as it holds all of the tubes in place.
So the easiest way to get this thing apart is remove the cover, remove the rear end, remove the housing with the reverser (valve block and all), and then tear into the reverser.
I added another pic showing the back half of the reverser inside the back of the housing. The tubes mostly run over to the valve block.
EDIT
I pulled it the rest of the way out this morning. Added 2 pics of the reverser out of the tractor.