  We investigate the strong Rayleigh property of matroids for which the basis
enumerating polynomial is invariant under a Young subgroup of the symmetric
group on the ground set. In general, the Grace-Walsh-Szeg\H{o} theorem can be
used to simplify the problem. When the Young subgroup has only two orbits, such
a matroid is strongly Rayleigh if and only if an associated univariate
polynomial has only real roots. When this polynomial is quadratic we get an
explicit structural criterion for the strong Rayleigh property. Finally, if one
of the orbits has rank two then the matroid is strongly Rayleigh if and only if
the Rayleigh difference of any two points on this line is in fact a sum of
squares.

Combinatorics