For years I have struggled with THAT! I don't think it does come apart. I have just spent the better part of the last 2 hrs rinsing, shaking, rinsing, etc I have tried draino and other solvents to loosen the gunk and put it in the dishwasher!! I had given up on it altogether for quite some time and used those softner BALL dispensers that float around BUT the machine dispenser works better. So, I have no recommendations but to SOAK it in hot water, SHAKE, Rinse, REPEAT till all or MOST is out.

PS throwing against the wall does NOT seem to help - now i have to look at a DIY wall repair site ;-) JK LOL

I have what sounds like a similar dispenser that clogged. It is one piece. I put about a tablespoon of baking soda in it, then dumped a few ounces of white vinegar and immediately pressed the palm of my hand firmly over the top opening to prevent air leakage. The reaction generates CO2 and builds pressure until it blows out some of the gunk. Then I used a dental pick to reach in from the bottom to clean out more gunk. A few rinses with *hot* water and it was good as new!