If you remove the matches from a matchbook, you now have a matchcover. Lots of collectors do this as the flat piece of cardboard is a lot easier to store and display. The object of most matchcover collectors seems to be to maximize the number of covers they can accumulate in their specialty and this makes it easier. Covers are also easier to mail as traders.

I don't have a real objection to collecting modern back-striker matches as covers. Value wise, the matchbooks are so common they will never be worth more than a few cents so there's no real concern about devaluing your collectible.

For forty years+ old back strikers, the situation is a little different. These are becoming scarce and I personally keep the matches in if it's a full book. For a partial book, I usually do strip the rest out and make it a cover.

If the back striker is a feature or printed stick, I always keep the matches in as the designer's intent would be lost otherwise. Also, if the casino cover is 1940's or earlier I keep any matches left in the matchbook intact as many of these are really scarce and do have some monetary value (in the $75-100 range in some cases).

You'll find differences of opinion on the "safety" issue of collecting books of "safety matches". I think the risk is very small and don't even consider it when deciding what to add to my casino match collection. There have been some ill-conceived regulations put forward recently in mailing safety matches or carrying them on an airplane. The risk is negligible, but the Feds seem to feel they must be doing something to further the quest of a risk-free society. You can mail safety matches; there's a strange regulation against mailing less than a full caddy of them (!). Collectors mail matches all the time. Risk is very small if you take reasonable precautions; keep the match heads from contacting the strikers and they are just sulfur on a stick and won't spontaneously burst into flame.

Both matchcovers and matchbooks will fit into the plastic pages made to hold such collections. There are separate size pockets in the pages for the narrow and wide matchbooks. Your collection of matchbooks will be thicker than a collection of matchcovers, so take that into account in providing shelf space. I have 25 or so binders for my casino match collection and would be half that if I shucked all the matchbooks. I like them as they were when used in the casino, though!