A notion of rigidity with respect to an arbitrary semidualizing complex C
over a commutative noetherian ring R is introduced and studied. One of the main
result characterizes C-rigid complexes. Specialized to the case when C is the
relative dualizing complex of a homomorphism of rings of finite Gorenstein
dimension, it leads to broad generalizations of theorems of Yekutieli and Zhang
concerning rigid dualizing complexes, in the sense of Van den Bergh. Along the
way, a number of new results concerning derived reflexivity with respect to C
are established. Noteworthy is the statement that derived C-reflexivity is a
local property; it implies that a finite R-module M has finite G-dimension over
R if it is locally of finite G-dimension.