  The off-shell version of the c-map is presented, based on a systematic
off-shell reduction from four to three space-time dimensions for supergravity
theories with eight supercharges. In the reduction, the R-symmetry group is
enhanced to local [SU(2) x SU(2)]/Z2 =SO(4) and the c-map is effected by a
parity transformation in the internal space that interchanges the two SU(2)
factors. Vector and tensor supermultiplets are each others conjugate under the
c-map and both can be dualized in three dimensions to (on-shell)
hypermultiplets. As shown in this paper the off-shell formulation indeed leads
to a clarification of many of the intricate issues that play a role in the
c-map. The results for off-shell Lagrangians quadratic in space-time
derivatives are analyzed in detail and compared to the literature. The
underlying reasons are identified why not all of the four-dimensional tensor
multiplet Lagrangians can be in the image of the c-map. The advantage of the
off-shell approach is, that it also enables a systematic analysis of theories
with higher-derivative couplings. This is demonstrated for a particular
subclass of such theories, which, under certain conditions, are consistent
under the c-map. In principle, explicit results for realistic four-dimensional
type-II string compactifications can be explored in this way.
