To preface, I live in a 12 unit condo building. What I am describing happens even if I am not home, and my neighbor (whose kitchen shares a wall with mine) is not home and no water is in use.
What would cause rancid/foul water to back up out my washing machine drain pipe (at the end of the kitchen)? This happens when the washing machine is not in use or any other water is not on. At the same time, I hear a gurgling like the sink is backing up even though the water is not on. This only happens on SUNDAY about once a month.

You share common drain pipe with other units above you. This drain is clogged someplace below your apartment. When upstairs apartments use water, water hits the blockage in the pipe, backs up, and overflows at the first possible opening - which happens to be your washer stand pipe.

You don't have to be home for the sewer back up to happen. It depends on the condos above you and their usage of plumbing fixtures

You should tell tenants in the units above not to use their washing machine and kitchen sink and have that pipe cleaned to prevent future back ups. Overflowing drains in lower units in multi-story apartment building is one of the disadvantages of living on the bottom floor

Hope that explains

Milo

Thank you so much! I own my unit, so the HOA is sending a plumber to check. I have no issue when I do laundry during the week, it doesn't back up! I hope that the clog/issue is in "common pipes" and not pipes exclusive to my unit so that I don't have to pay for it.