We describe methods for implementing separation algorithms for domino-parity inequalities for the symmetric traveling salesman problem. These inequalities were introduced by Letchford (2000), who showed that the separation problem can be solved in polynomial time when the support graph of the LP solution is planar. In our study we deal with the problem of how to use this algorithm in the general (non-planar) case, continuing work of Boyd et al. (2001). Our implementation includes pruning methods to restrict the search for dominoes, a parallelization of the main domino-building step, heuristics to obtain planar-support graphs, a safe shrinking routine, a random-walk heuristic to extract additional violated constraints, and a tightening procedure to modify existing inequalities as the LP solution changes. We report computational results showing the strength of the new routines, including the optimal solution of a 33,810-city instance from the TSPLIB.