You my friend are a hero. Every search for this horribly annoying problem brought back the same old things to check. None of them were the answer. This is. This fixes it. I did this on two of my machines and they now behave perfectly. Thank you. Thank you. Thank you.
Desktop icons that rearrange themselves on reboot or a refresh or when a program starts seems quite common.
The usual workaround is to use microsoft's layout.exe or a third party program (eg Iconoid) to save and restore icon positions but this is not a solution.
After much time I tracked the problem down to a registry with corrupted 'permissions':
HKEY_CURRENT_USER\Software\Microsoft\Windows\Shell\Bags\1\Desktop
I could not get into Bags\1 - 'Error on opening' message but I could get into permissions which I reset including the 'owner' tab.
I imported a copy of another computers bags\1 settings although it had no similarity regarding icon settings. Made sure the icons were how I wanted them positioned and rebooted. Problem solved.
I hope this posting helps others with this tedious problem.Desktop icons that rearrange themselves on reboot or a refresh or when a program starts seems quite common.
The usual workaround is to use microsoft's layout.exe or a third party program (eg Iconoid) to save and restore icon positions but this is not a solution.
After much time I tracked the problem down to a registry with corrupted 'permissions':
HKEY_CURRENT_USER\Software\Microsoft\Windows\Shell\Bags\1\Desktop
I could not get into Bags\1 - 'Error on opening' message but I could get into permissions which I reset including the 'owner' tab.
I imported a copy of another computers bags\1 settings although it had no similarity regarding icon settings. Made sure the icons were how I wanted them positioned and rebooted. Problem solved.
I hope this posting helps others with this tedious problem.

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic