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](http://mathoverflow.net/a/135471/6094).
Sets with nonunique reconstructions are called *homometric*. Here is a snippet from the above paper, addressing a property of regular polygons:
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<img src="https://i.sstatic.net/XBNGN.png" width="400" />
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(From p.335 of conference version.)
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  [1]: https://i.sstatic.net/XBNGN.png