There are $n$ points on the unit circle,
$\pmb{r}_{j} = (\cos[\theta_{j}], \sin[\theta_{j}])$.
For every pairs of points ($j$, $k$), the following relation is satisfied,
$\pmb{r}_{j} \cdot \pmb{r}_{k} = \pmb{r}_{\sigma_{1}(j, k)} \cdot \pmb{r}_{\sigma_{2}(j, k)}$,
where
$[\sigma_{1}(j, k), \, \sigma_{2}(j, k)] \ne [j, k]$
$[\sigma_{1}(j, k), \, \sigma_{2}(j, k)] \ne [k, j]$.
In other words, there is no unique angle amongst the ${n \choose 2}$ angles subtended by pairs of points and the origin. What does this imply about the symmetry of the points? $n$ points possessing some rotational symmetry satisfy the constraints. Is rotational symmetry necessary for the constraints to be satisfied?
Any advice on how to tackle this problem would be appreciated.