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Automating garment manipulation is challenging due to extremely high variability in object configurations. To reduce this intrinsic variation, we introduce the task of "canonicalized-alignment" that simplifies downstream applications by reducing the possible garment configurations. This task can be considered as "cloth state funnel" that manipulates arbitrarily configured clothing items into a predefined deformable configuration (i.e. canonicalization) at an appropriate rigid pose (i.e. alignment). In the end, the cloth items will result in a compact set of structured and highly visible configurations - which are desirable for downstream manipulation skills. To enable this task, we propose a novel canonicalized-alignment objective that effectively guides learning to avoid adverse local minima during learning. Using this objective, we learn a multi-arm, multi-primitive policy that strategically chooses between dynamic flings and quasi-static pick and place actions to achieve efficient canonicalized-alignment. We evaluate this approach on a real-world ironing and folding system that relies on this learned policy as the common first step. Empirically, we demonstrate that our task-agnostic canonicalized-alignment can enable even simple manually-designed policies to work well where they were previously inadequate, thus bridging the gap between automated non-deformable manufacturing and deformable manipulation. Code and qualitative visualizations are available at https://clothfunnels.cs.columbia.edu/. Video can be found at https://www.youtube.com/watch?v=TkUn0b7mbj0.