There are two major uses of abstraction in planning and search: refinement (where abstract solutions are extended into concrete solutions) and heuristics (where abstract solutions are used to compute heuristics for the original search space). These two approaches are usually viewed as unrelated in the literature. It is reasonable to believe, though, that they are related, since they are both intrinsically based on the structure of abstract search spaces. We take the first steps towards formally investigating their relationships, employing our recently introduced framework for analysing and comparing abstraction methods. By adding some mechanisms for expressing metric properties, we can capture concepts like admissibility and consistency of heuristics. We present an extensive study of how such metric properties relate to the properties in the original framework, revealing a number of connections between the refinement and heuristic approaches. This also provides new insights into, for example, Valtorta's theorem and spurious states.