While leaving it alone is the default option, when I visited Kitt Peak not long ago, the guide said they used Orvis Horse Shampoo for cleaning the mirror on the 2.1m scope. Dunno if this was a joke, but the guide seemed serious. He also said they once found some footprints on the mirror and discovered that a ring-tailed
cat had used the mirror for a dining table.
Having worked in a military camera repair shop where we repaired very high-end aerial cameras, I say: proceed with caution. If you really want to clean a mirror, here is one technique:
http://www.rfroyce.com/cleanfsm.htm. I'd be nervous about this as the first wash water could be contaminated with grit. I'd use distilled water all the way through the process. The writer didn't mention getting all the gritty stuff off first. You don't want to rub the grit into the coating. Blow it off with an ear syringe (from a pharmacy) and pick off stubborn bits with a little sable paintbrush from an art supply store (and don't rub the brush on your skin — keep it clean). Then wash and rinse. While I haven't tried this, Kodak Photo Flo mixed in distilled water to a to about half the strength used for film might be a good final rinse solution that wouldn't leave water spots. We always had good results with it on film.
I just cleaned some sticky goo from rotted foam off the reflex mirror in an old Nikon F. I used lacquer thinner and a handful of cotton swabs. The dust is first blown off with clean compressed air. Then a swab damp with thinner is rolled on the mirror such that the dirt it picks up is pulled up away from the mirror surface. The swab gets only one roll and then is tossed and a new one is used for the next bit. This takes patience but you can prevail if you're careful.
