This is, in general, a difficult problem, addressed in the paper below. It goes under the phrase: the beltway reconstruction problem.
Lemke, Paul, Steven S. Skiena, and Warren D. Smith. "Reconstructing sets from interpoint distances." Discrete & Computational Geometry. Springer Berlin Heidelberg, 2003. 597-631.
There is some info in an earlier MO question. Sets with nonunique reconstructions are called homometric. Here is a snippet from the above paper, addressing a property of regular polygons:
(From p.335 of conference version.)
