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When Can Coins Be Cleaned?
"Never clean coins" is the simple, oft-repeated message, often said because the true answer - "It depends" - is not only ambiguous, but encourages people who don't know what they're doing to try it out, potentially ruining an interesting and/or valuable coin. Because people simply aren't very good at telling the difference between "that's just dirt and will come right off" and "that's chemically bonded with the coin's surface and is part of the coin now and won't come off without harsh cleaning which will damage the coin further".
But to answer the OP's specific question given in the thread title: Coins not only can, but should, be cleaned in the following circumstances:
- Coins dug up from the ground. These usually have literal, physical dirt still attached to them, and the dirt should be washed off. This includes every single ancient and mediaeval coin currently in existence. Ancient bronze and copper coins that have been buried in the ground for 2500 years are well on their way towards turning back into the copper ore from whence it originally came; they come up out of the ground looking like little green rocks, and the boundary-line between "dirt" and "coin's surface" is fuzzy and indistinct; the process by which such coins are cleaned is more an art than a science, and it is very easy to ruin such a coin forever with "improper cleaning". And within ancient coin collectors there's a school of thought that says a coin should only be cleaned up to the point where it can be fully identified, then the cleaning should stop, to try to preserve the coin as best as possible in its "as-found" state. Soap and water? Possibly effective for a modern coin, otherwise, no.
- Coins with active corrosion. "Bronze disease" is a contagious form of corrosion that can be found on copper and bronze coins. Usually pale-green and powdery, if left unchecked (and especially in warm damp conditions) it will continue to spread until the entire coin becomes a crumbly mess. Such corrosion must be treated to ensure it does not return at a future date. Again, improper cleaning can not only destroy a coin that didn't really need cleaning in the first place, but can fail to stop or can even encourage the contagious forms of corrosion. Rust on iron, zinc rot and aluminium cancer are similar forms of corrosion on other metals. Soap and water would be totally ineffective, and would even make some kinds of corrosion (such as aluminium cancer) worse.
- Coins damaged as a result of improper storage. "PVC damage" is the most famous example of this. Cleaning is advised in such situations because the "goo" left behind by degraded PVC can continue to eat into a coin's surface if not removed, and can damage or destroy whatever new holder or album the coin is placed in. Soap and water would be largely ineffective.
- Coins with "foreign matter" stuck to them. Paint, glue, varnish, stickytape residue, partially decomposed rubber bands and such like. Such substances have not reacted chemically with the metal and metal compounds of the coin's surface. You will usually want to use some kind of solvent: water, ethanol, acetone, xylene, hexane is the "polarity ladder" of solvents; work through them in that order and one of them will usually work. Acetone is a good "middle of the road" solvent, able to remove most forms of "goo" you're likely to find on a coin, plus it evaporates away quite readily, leaving no residue behind. Soap and water might work, but probably won't.
- War medals. Nobody can stop an old soldier from polishing up their medals, so we don't bother trying. War medals are one of the few areas of numismatics where giving an item a good thorough polishing does not hurt the collector value.
Coins that are heavily corroded or badly affected by non-contagious corrosion are generally considered "non-collectable" by coin collectors. They are un-salvageable, as the corrosion has almost certainly damaged the surfaces; removing the corrosion chemically (eg with acid or ammonia) would likely leave behind a cratered moonscape; unless such a coin is super-rare, nobody would want it and they are just worth scrap metal value. So yes, you can clean them, but no, you're not likely to "make the coin better" by doing so. And soap and water will do nothing in such cases.
Coins that are merely oxidized, having turned black on exposure to a high-sulfur atmosphere, are usually deemed "perfectly fine" by collectors and not in need of cleaning. Soap and water won't do anything to remove a coin's patina. A coin stripped of it's patina doesn't look "shiny and new again", it looks dull and bland, because the original shininess, or lustre, of a coin is created by the minting process and can't be put back by cleaning or polishing. It is possible to attempt to re-create artificial shininess on a dull coin by extra-harsh polishing, a process known in coin collecting as "whizzing"; this practice is universally condemned as it damages a coin's surface.
Finally, a quick addendum on the "water" mentioned above: most people's tap water contains chemicals (chlorine, salts, etc) that can damage coins if left to soak in it for prolonged periods, and rain water is usually slightly acidic. So we usually recommend using distilled or deionized water instead.
TLDR: We say "don't clean coins", not because coins should never ever be cleaned under any circumstances, but because it's actually really hard to clean coins properly, and really easy to clean coins badly. And "soap and water" is almost always in that latter category.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis