How To Align Rear Lsd Side Gears Without Disassembly

Hello. I have an 8" rear with Trac Lok that I'm putting in my 1980 Pinto. I recently took the axles out to be redrilled to a 5 x 4.5 bolt pattern. I have no idea how, but the teeth on the D/S side gear and the clutch hub became misaligned (see photos), which is preventing me from reinstalling the axle fully. There is only one toothed part on the passenger side, and that axle went in with no issues.

I feel that I am missing something painfully obvious. Maybe that's because I don't fully understand how an LSD works, but how can I turn one toothed part (either the side gear or the hub), without the whole assembly turning on me? With the P/S axle, wheel, and tire installed and chocked, I tried inserting the D/S axle as far as it would go and turning, but the damn thing doesn't want to budge that way. I really do not want to disassemble the diff if I can help it.

Thanks in advance from this noob...

Thank you, horse sence! I was on the right path, but your reply gave me the motivation to keep turning that axle, and I was able to get it to seat properly in the diff/housing. The tire was giving me too much leverage (read: it was spinning the passenger tire, even though it was chocked,) so I removed it, put on some heavy gloves, and turned the axle flange by hand. Here's a picture of the final result: