For matroids M and N on disjoint sets S and T, a semidirect sum of M and N is any matroid K on $${S \cup T}$$ that, like the direct sum and the free product, has the restriction $${K|S}$$ equal to M and the contraction K/S equal to N. We abstract a matrix construction to get a general matroid construction: the matroid union of any rank-preserving extension of M on the set $${S \cup T}$$ with the direct sum of N and the rank-0 matroid on S is a semidirect sum of M and N. We study principal sums in depth; these are such matroid unions where the extension of M has each element of T added either as a loop or freely on a fixed flat of M. A second construction of semidirect sums, defined by a Higgs lift, also specializes to principal sums. We also explore what can be deduced if M and N, or certain of their semidirect sums, are transversal or fundamental transversal matroids.