I work in the appliance manufacturing industry. Most dishwasher tubs are made of a plastic usually polypropylene . Very inert material. Detergent used in dishwashers is very strong alkaline, water is at or above 120F. Any Styrene, ABS, polycarbonate and blends are at high risk. I think Paterson uses some form of acetal/nylon blend. Pretty tough stuff. Still I wouldn't subject anything I care about to a dishwasher chemicals. It will literally etch glass.
I always dip my reels with film into photo-flo in DI water. If they get gunked up I will throw them away and buy new. I don't want to scratch film by trying to avoid final rinse with color or photo-flo with black and white. I wash them immediately with very hot water. I've been pretty lucky so far.
I have pretty much adopted Jobo since I got a deal on a CPP2. All in all the Paterson reels (IMHO) are so much better in every way. Sad they won't work with Jobo tanks.
It's still hard to beat System 4 tanks for easy no drama in the dark developing.
Best Regards, Mike