Blocking constraints are ubiquitous in machine scheduling. In a flowshop scenario, a machine might get blocked by a job that has just been processed by the same machine. This happens when there is no buffer where the job can temporarily go before the next machine is available to process it. This also happens when two successive machines depend on a mutually exclusive resource to process the same job. In this paper, we show that under certain conditions, two successive machines operating under blocking constraints involving mutually exclusive resources can virtually be treated as a joint machine. Before performing search, we then use this makespan preserving property in reengineering a permutation flowshop. The reengineered flowshop model can actually be used by any arbitrary search algorithm to improve efficiency. We show an example of machine conjoining in the waste treatment industry. We also empirically show that such reengineering helps state-of-the-art flowshop search algorithms obtain significantly better makespan values than when the original model is used. Flowshop reengineering is appealing particularly because it helps obtain a higher abstraction level and any search algorithm can use it as a preprocessing step.