In online minimum cost matching on the line, $n$ requests appear one by one and have to be matched immediately and irrevocably to a given set of servers, all on the real line. The goal is to minimize the sum of distances from the requests to their respective servers. Despite all research efforts, it remains an intriguing open question whether there exists an $O(1)$-competitive algorithm. The best known online algorithm by Raghvendra [SoCG18] achieves a competitive factor of $Θ(\log n)$. This result matches a lower bound of $Ω(\log n)$ [Latin18] that holds for a quite large class of online algorithms, including all deterministic algorithms in the literature. In this work we approach the problem in a recourse model where we allow to revoke online decisions to some extent. We show an $O(1)$-competitive algorithm for online matching on the line that uses at most $O(n\log n)$ reassignments. This is the first non-trivial result for min-cost bipartite matching with recourse. For so-called alternating instances, with no more than one request between two servers, we obtain a near-optimal result. We give a $(1+\varepsilon)$-competitive algorithm that reassigns any request at most $O(\varepsilon^{-1.001})$ times. This special case is interesting as the aforementioned quite general lower bound $Ω(\log n)$ holds for such instances.