How to fix a dishwasher that won't drain - look up!

We recently had a problem with our dishwasher -- it wouldn't drain. Merely scraping stuff off the bottom of the washer didn't help. I pulled the rotor out, and took off a housing above the motor, but this didn't reveal lots of gunk either.

I tried running with white vinegar and baking soda. I think it cleaned things up, but still water at the bottom.

Next I took the baseboard off and unconnected the drain tubing. This allowed water to drain freely, so it looked like things were draining from the washer fine.

At this point I gave up. I saw the drain pipe from the bottom of the washer snaking up to the adjacent sink, but I thought it'd just drain easily into a big pipe up in the sink.

But then we discovered that there's actually a choke-point up high when the drain pipe empties into the adjacent sink. There's a junction, which apparently also serves as an "air gap". This air gap protrudes up out of your sink. I thought it was a remnant of a soap dispenser or a pull-out sprayer, but it served as a good choke point for gunk to build up.

After cleaning that, the washer drained fine.