Climate control blower intermittent failure

Sometimes when I start the engine, the air conditioner/heater blower does not come on. No combination of dashboard settings will coax the blower into working. You can feel the presence of cold air inside the vent outlet, but the air isn't moving at all.
When this happens, restarting the engine and turning the AC system on/off has no effect.
This only happens randomly, the rest of the time, it works perfectly.
I have been able to get the blower working every time, though. Here is what works: Banging on the dashboard, repeatedly slamming the driver's side door very hard, and going over bumps in the road.
The door slamming seems to be the most reliable method.
Once the system is working, it will not fail again until the engine is started randomly at some point. It will not fail once it is going. It will not suddenly turn off while driving, but it will suddenly start working.
I have taken it in to a shop, but they could not do anything with it because it would not fail for them.
Could something be loose? Why will it start up randomly but never randomly shut off? What is the connection to the driver's side door? Is something near it in the engine that would be affected by a door slamming?
Thanks for any insight you may have.