The best double-sided-view coin mount I've seen is made with Mylar/Melinex. Relatively easy, completely non-invasive, inert, and it shows the whole coin attractively -- even if other kinds of things are in the same frame. Here's how:
1. Cut two mats (face and back) with identical opening patterns. Since most CMC's won't cut circles small enough for most coins, I use hexagonal or octagonal openings, which can be cut dime-size with no problems. The openings should fit the coins not-too-snugly.
Hexagons or octagons may be cut on any straight cutter, but positioning the cuts on a mirror-image mat for the back view is tricky. For jobs like this, if you don't have a CMC, I suggest you buddy up to a framer who does.
I like to cut the front mat with reverse bevels, and the back mat with bevels showing, but the opposite works as well. Because the bevels will be seen, I suggest using black-core or color-core mats -- unless your mats will be white, of course.
2. Cut one or more middle mats, to equal the thickness of the coin. Opening pattern is same as front & back mats. BEFORE CUTTING the middle mat(s) apply strips of 3M #889 double-sided tape to cover the entire surface of both sides of the board. Leave the release papers intact during cutting. Do not use ATG.
To assemble:
3. Remove release papers from the tape strips on one side of the middle mat(s) and lay on a clean sheet of Mylar/Melinex. Burnish thoroughly.
4. Turn the middle mat film-side down. Lay coins in place, being careful to rotate them so their fronts are in proper position.
5. Peel release papers from that side and lay on the other piece of Mylar/Melinex. Burnish thoroughly.
6. Add front & back mats to the assembly, and fit as usual.
Note that if several coins of different thicknesses are mounted together, some may bulge the mounts and some may be loose. Don't worry about slightly bulging mount, but looseness could result in the coin turing in its mount, during typical consumer transport & handling.
If you want a photo or two, let me know your email address.