Tree reconciliation is the mathematical tool that is used to investigate the coevolution of organisms, such as hosts and parasites. A common approach to tree reconciliation involves specifying a model that assigns costs to certain events, such as cospeciation, and then tries to find a mapping between two specified phylogenetic trees which minimizes the total cost of the implied events. For such models, it has been shown that there may be a huge number of optimal solutions, or at least solutions that are close to optimal. It is therefore of interest to be able to systematically compare and visualize whole collections of reconciliations between a specified pair of trees. In this article, we consider various metrics on the set of all possible reconciliations between a pair of trees, some that have been defined before but also new metrics that we shall propose. We show that the diameter for the resulting spaces of reconciliations can in some cases be determined theoretically, information that we use to normalize and compare properties of the metrics. We also implement the metrics and compare their behavior on several host parasite data sets, including the shapes of their distributions. In addition, we show that in combination with multidimensional scaling, the metrics can be useful for visualizing large collections of reconciliations, much in the same way as phylogenetic tree metrics can be used to explore collections of phylogenetic trees. Implementations of the metrics can be downloaded from: https://team.inria.fr/erable/en/team-members/blerina-sinaimeri/reconciliation-distances/.