We investigate, in a variational setting, the relationship between the Gromoll-Meyer pairs of a dynamically isolated critical set and the Conley index pairs of its isolating invariant neighbourhoods. We show that the information given by the critical groups of such a set is equivalent to that given by the Conley index. This allows us to derive - in a non-compact setting - various invariance properties for the Conley index from those of the critical groups, as well as a formula relating the degree of a gradient vector field in an isolating neighbourhood to the Conley index pair associated with it.