Odd Steinhaus problem for finite sets

Call a finite subset $S$ of the plane with an even number of points an odd Jackson set, if there is an $Asubset mathbb R^2$ such that $A$ meets every congruent copy of $S$ in an odd number of points.

Are there any odd Jackson sets?

For a bit of history of related questions (and an explanation of the nomenclature), see here.
For my motivation, see here.