It seemed as if one solenoid was engaging but one did not. We heard no click coming from the brake drum.
Today, Gerald and I pulled the drum from the suspect wheel aided by his 3-ton hydraulic jack. Gerald doesn't just have tools; he's got a 3-ton hydraulic jack. How many folks have one of these industrial grade contraptions lying around in the garage.
Back to the brake. Take a close look at this picture and you won't see a solenoid. However, that circular piece of iron toward the bottom is an electromagnet. Mystery solved. Apparently, the electromagnet rides on the inside of the drum held there by a spring so when the current is applied to it, it really doesn't have to travel any distance at all so there is no thump or click.
This makes me wonder about what is happening in the other wheel. Perhaps the spring is missing or too weak to keep the electromagnet pushed gently against the hub. We'll pull that wheel later and take a look.
We went looking for a defective solenoid but found an electromagnet. But, we should have been looking for an electromagnet, because a solenoid is really an electromagnet.