Where clogs hide
Almost always at a bend. The top elbow where the gutter feeds the pipe, and the bottom elbow where it kicks out at the ground, catch debris like a drain trap. Straight vertical runs rarely clog on their own. This is why blowing out the gutter trough and calling it done fixes nothing.
How a pro clears one
Flush from the top with a hose at full flow. If it backs up, work the clog with a plumber snake or disassemble the elbow (they are usually screwed, not welded). A crew that flushes every downspout and shows you the water running clear at the bottom has actually finished the job.
The 30-second test you can do yourself
During the next decent rain, walk the house and watch each downspout outlet. Strong steady flow is a pass. A trickle while the gutter above overflows is a clog. No tools, no ladder, and now you know exactly what to tell the crew.