why would my chain slip on the road, but not when the bike is on a stand? The chain on my bike has started to slip a lot -- every couple of seconds when riding. It happens, as far as I can tell, in every gear, and is unrelated to shifting.

The precise nature of the problem is hard to diagnose because I can not make it happen when the bike is on a stand, it happens only on the road.

What kind of chain/derailleur problem would manifest only on the road and never when the bike is on a stand?
You likely have a problem with worn drivetrain components. You only really notice the jumping under load is my experience. 

It's either a worn chain or a worn cassette in most cases. If you replace the cassette, replace the chain too.

Surprisingly enough, a worn chain and a worn cassette may work well together till one of the two parts are replaced.