Of 14 essentially identical machines, 6 could see each other and the outside world, 8 couldn't even see other machines on the same switch.
I resorted to randomly rebooting stuff to try to get them to reset state, renegotiate the links and so on. Nothing worked.
Experimentally, I moved some of the non-working systems over to the old unmanaged switch and connected that switch to the new one, and they were all visible.
I then decided to hit the reset button on the switch, and it came up with faults after a reset.
In the end I moved everything back to our old unmanaged switch, it was all just plug-and-play and worked.
Ho-hum. Now to try to get the serial console to work to see if the 2824 will tell me what's going wrong.