AoPS Solution

There are \(10!\) ways to put the shells in the sand, not considering rotations and reflections. Arrangements can be reflected or not reflected and can be rotated by 0, 1/5, 2/5, 3/5, or 4/5, so they come in groups of ten equivalent arrangements. Correcting for the symmetries, we find that there are \(\frac{10!}{10}\)\(=\boxed{3628800}\) distinct arrangements.