I finally did a session with tech support when the same problem cropped up again. The tech, as usual, was great. (It always amazes me that they are able to learn packages a convoluted as ESRI's as well as they do.) We were able to narrow the problem down to a software problem with the particular map app I was using. It was the "Basic Map Viewer", which you get to by clicking Share, Create A Web Map, Configurable Apps, then the default Basic Viewer.
The tech was willing to keep at the problem to the bitter end, but mentioned that the Web AppBuilder had a better track record of reliability than the Configurable Apps that I had used. I tried that. The WebAppBuilder let me recreate my app quickly and easily. When in doubt, refresh, reboot or rework. It can be worth a pound of cure.