How to unclog the gutter downspouts from the ground

There is an adapter that connects the output of a leaf blower to the bottom of the downspout and blows the debris back out the way it came (sometimes you can even do it without an adapter). It works pretty well if it is a recent accumulation. Note, though, that you will get covered in a shower of disgusting stuff (gravity working for you).

But if it has been decomposing for several seasons and turning into a solid block of adobe filling the inside the downspout, that may not work. In that case, the snake and hose is probably the tool of choice to try first. Smacking the downspout in the vicinity of the clog can help loosen it.

They make a special nozzle for the hose that puts out both a high pressure forward spray, and side sprays that also spin the nozzle, creating a rotating jet of water that helps to break things up and wash it out. It can be a long slow process that will raise your water bill and flood your yard, turning it into a mud field by the time you're done.

Sometimes the fastest solution to stubborn downspouts is just to start taking sections apart. A few screws and a section comes off. Clogs tend to build from the bottom up so if you're lucky, the bottom-most section will either do it, or make it easier to get the rest out by simpler means. Once a section is removed, low tech smacking, blowing, or pushing is usually enough to clean it.