  We characterize the combinatorial types of symmetric frameworks in the plane
that are minimally generically symmetry-forced infinitesimally rigid when the
symmetry group consists of rotations and translations. Along the way, we use
tropical geometry to show how a construction of Edmonds that associates a
matroid to a submodular function can be used to give a description of the
algebraic matroid of a Hadamard product of two linear spaces in terms of the
matroids of each linear space. This leads to new, short, proofs of Laman's
theorem, and a theorem of Jord{\'a}n, Kaszanitzky, and Tanigawa, and Malestein
and Theran characterizing the minimally generically symmetry-forced rigid
graphs in the plane when the symmetry group contains only rotations.
