If objects, such as the atmosphere, that are at rest relative to the Earth appear contracted to the muon, then objects moving with the muon must appear shortened by the same factor when viewed from the Earth. Consider for example the size of the muon itself. Muons are very tiny particles and have virtually no extent, but let us for the sake of illustration assume that as measured in the muon's frame of reference, it has a width of one centimeter. Thus, as expected, lengths of objects attached to the muon appear contracted as seen from Earth, whereas lengths of objects moving with the Earth appear contracted as measured in the muon's frame. One last thing to emphasize is that the entire discussion so far refers only to one dimensional motion.
Length contraction applies only to lengths along the direction of motion.
It is of course possible to apply the postulates of Special Relativity to three dimensional motion, but the analysis gets considerably more complicated, and no essential new features emerge.