The well-known graphical calculus for spherical fusion categories is extended to arbitrary monoidal categories and is developed in detail for fusion categories without assumption of sphericity. This treatment makes a systematic connection to the underlying category, and attributes traditionally associated to spherical categories are found to exist independently of any spherical structure. This leads to elementary proofs some previously non-elementary results, including Ocneanu rigidity and the fact that the quadruple-dual functor is isomorphic to the identity functor. In addition some new technical tricks for the efficient classification of fusion categories for a given set of fusion rules become available.

The calculus is used to make some remarks on the conjectural equality between the Turaev-Viro invariants given by a spherical fusion category [special characters omitted] and Reshetikhin-Turaev invariants given by the (modular) quantum double D([special characters omitted]). A version of the fact that sphericity of [special characters omitted] implies modularity of D([special characters omitted]) holds even without sphericity. A gap in existing invariance proofs for the Turaev-Viro invariants of spherical categories is described, and is filled for the case of unimodal categories. It is argued that there is no reason to expect invariance for non-unimodal categories. Non-invariance would cause the conjectural equality to fail on a technicality because the Reshetikhin-Turaev invariants of quantum doubles and the Turaev-Viro invariants would have different domains of definition.