  Modulo trivial exceptions, we show that smoothly nontrivial symplectic sums
of symplectic 4-manifolds along surfaces of positive genus are never rational
or ruled, and we enumerate each case in which they have Kodaira dimension zero
(i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In
particular, a symplectic four-manifold of Kodaira dimension zero arises by such
a surgery only if it is diffeomorphic to a blowup either of the K3 surface, the
Enriques surface, or a member of a particular family of T^2-bundles over T^2
each having b_1=2.
