This paper discusses the role of symmetry and transformations in the analysis of coherent structures in turbulent flows. It focuses on how symmetry considerations can extend the available data and simplify the analysis of turbulent flows. The paper presents a detailed analysis of several standard geometries, including plane Poiseuille flow, Poiseuille flow in a rectangular channel, and the Bénard problem (convection), as well as other geometries such as flow past bodies of revolution, flow in a circular pipe, flow past a circular cylinder, and plane Couette flow. For each of these geometries, the paper discusses the symmetry properties and how they can be used to generate admissible flows and to simplify the analysis of the flow. It also discusses the use of symmetry to reduce the number of basis functions needed for the analysis, and how this can be applied to transform eigenfunctions or coherent structures to other geometries. The paper also discusses the application of symmetry in the context of the Galerkin procedure for general flows, and how symmetry can be used to zero out certain quantities that are not expected to be present in the flow. The paper concludes with a discussion of the implications of these symmetry considerations for the analysis of turbulent flows, and how they can be used to improve the accuracy and efficiency of the analysis. It also discusses the use of symmetry in the context of the continuity equation for incompressible flows, and how this can be used to reduce the number of components needed to describe the flow. The paper emphasizes the importance of symmetry in the analysis of turbulent flows and how it can be used to simplify the analysis and improve the accuracy of the results.This paper discusses the role of symmetry and transformations in the analysis of coherent structures in turbulent flows. It focuses on how symmetry considerations can extend the available data and simplify the analysis of turbulent flows. The paper presents a detailed analysis of several standard geometries, including plane Poiseuille flow, Poiseuille flow in a rectangular channel, and the Bénard problem (convection), as well as other geometries such as flow past bodies of revolution, flow in a circular pipe, flow past a circular cylinder, and plane Couette flow. For each of these geometries, the paper discusses the symmetry properties and how they can be used to generate admissible flows and to simplify the analysis of the flow. It also discusses the use of symmetry to reduce the number of basis functions needed for the analysis, and how this can be applied to transform eigenfunctions or coherent structures to other geometries. The paper also discusses the application of symmetry in the context of the Galerkin procedure for general flows, and how symmetry can be used to zero out certain quantities that are not expected to be present in the flow. The paper concludes with a discussion of the implications of these symmetry considerations for the analysis of turbulent flows, and how they can be used to improve the accuracy and efficiency of the analysis. It also discusses the use of symmetry in the context of the continuity equation for incompressible flows, and how this can be used to reduce the number of components needed to describe the flow. The paper emphasizes the importance of symmetry in the analysis of turbulent flows and how it can be used to simplify the analysis and improve the accuracy of the results.