a free spectrahedron d is the matricial solution set of a linear matrix inequality (lmi). thus , for some positive integer g, d is a subset of the union, over n, of g-tuples of n by n matrices. free spectrahedra arise naturally, in several contexts including model engineering problems and the the theory of operator systems and completely positive maps. we consider the problem of classifying, up to affine linear equivalence, free bianalytic mappings from one free spectrahedra d to another e. under some irreducibility inspired hypotheses on d, there are few such maps, d must support an underlying algebra/module structure and the map itself has a particularly pleasing form arising from this algebra/module structure. the work is joint with meric augat, bill helton and igor klep.