Rational quartic spectrahedra in real 3-space are semialgebraic convex subsets of semidefinite 4×4 real symmetric matrices, whose boundary admits a rational parameterization. The Zariski closure in complex projective space of the boundary is a symmetroid. If the symmetroid have only simple double points as singularities, it is irrational, in fact birational to a K3-surface, so rational symmetries are special. Rational quartic symmetroids have a a triple point, an elliptic double point or is singular along a curve. Reporting on common work in progress with Martin Helsøe, I shall give several examples and first results towards a classification.