Distinct Point Sets With Same Distances

A given configuration of n points in space uniquely determines 
n(n-1)/2 point-to-point separations, but the converse is not true.  
Given a set of n(n-1)/2 separations there may be more than one 
configuration of n points with those separations.  For example, 
given the set of 10 separations

 1,  1,  1,  1,  sqrt(2),  sqrt(2),  2,  2,  sqrt(5),  sqrt(5)

we can construct either of the two 5-point configurations shown below 

                b c d                   a b c
                a   e                     d
                                          e

Can anyone provide other examples of such "equivalent configurations", 
i.e., two or more distinct configurations of points (not counting 
rotations and reflections) sharing the same set of point-to-point 
separations?  It's also possible to construct one-dimensional examples.
as discussed in Isospectral Sets In One Dimension.

I think the following THREE distinct configurations of eight points 
each have the same set of 28 point-to-point separations:

   * * * *            * * *             * *   * 
   * *                    *               *
   *                      * *             * *
       *                * *               * *

Of the 12870 possible arrangements of eight points on a 4x4 grid, I 
think there are only 1120 different sets of separations.  Of course, 
a lot of this reduction is due to rotations and reflections, but not 
all.  It appears that such configurations are not uncommon.