wow...I have only seen one C-210 with the rear gear doors still installed. Have you tried
www.cessnaowner.org .??
The landing gear is an engine driven hydraulic system with a lot of limit switches for sequencing the doors and gear swing. The limit switches operate solenoids in the hydraulic pack, the sequence is doors open, gear down or up, doors close, selector handle returns to neutral. This requires limit switches on the three sets of doors for both open and close as well as up lock and down lock for the gear. in 1979 (I think) Cessna removed the doors and eliminated the solenoids, and also I think there is an STC to remove the doors.
Because some shops and most pilots really don't understand the system, owners have paid for needless repairs, power pack overhauls, and door-removal modifications. Actually, there is an electrical circuit involving the up-lock and down lock microswitches along with the handle switch that energizes the door solenoid valve to the door-close position. If your doors won't close on either the gear-up or gear-down cycle, then the problem could be in this circuit. A simple cleaning of the microswitches may fix the problem, or maybe a broken wire that the broken area re-connects when the weight of the plane is on the gear.
I am not a mechanic, but I used to fly C-210s carrying canceled checks. These are a couple of things I learned from experience and the night mechanic. Hopefully soon someone with more knowledge on Cessna landing gear will see this and give us more information.
And welcome to the site.!!