Not in the sense of a mirror, but yes, it does, nondirectionally. More importanty, it tends to radiate at frequencies not absorbed by the material.
Look, "reflection" is a specific concept, OK? It's a different thing than absorption and emission. They are two different effects, which is why we use different words to describe them. There is some refraction by clouds, and I'm sure some trivial percentage is refracted multiple times to end up headed back towards the surface, but that's about it.
Why does the distinction matter? Actual reflection of IR nearly blocks radiative cooling (at some point the reflective surface, not being perfect, heats up and starts radiating). This is why a really good thermos needs a reflective layer in addition to a vacuum gap. If IR were being reflected close to the emitter, which it's not, changing the frequency to one not reflected would make a huge difference.
Absorption is very different. Sure, the atmosphere absorbs IR and becomes warmer as a result, but it's not like that happens meaningfully within a few yards of the ground! Changing the frequency of your thermal radiation a bit will not make a meaningful difference in cooling. OTOH, covering something sitting in the sun with a reflective surface makes a significant difference.