The present work revisits hydrodynamic linear stability theory from the perspective of Lie symmetries. For the linearized Navier-Stokes equations of a two-component shear flow, the expected symmetries and equivalence transformations of the Navier-Stokes group are recovered, while additional ones arise due to linearization. The concept of symmetries and equivalence transformations in space and time is extended to spectral space, where the new definition of a spectral equivalence transformation is introduced. These are not fundamentally new transformations, as they correspond to space-and-time symmetries or equivalence transformations; however, they take on a different form that is used throughout the subsequent analysis. In particular, the Squire transformation is derived from such a symmetry generator of a spectral equivalence transformation with regard to the coupled Orr-Sommerfeld and Squire system, including the vorticity. As another step, these spectral generators are applied to an adjoint-projected residual functional. The variation of this functional is studied via the variation of the differential operator. This approach yields algebraic invariance conditions as well as derivative expressions. At a direct-adjoint eigenpair corresponding to an isolated eigenvalue, these derivatives determine the derivatives of the eigenvalue branch through the standard adjoint sensitivity relation and thus the group velocity. One specific result thereof is that the homogeneous Squire equation does not admit stationary-frame double saddle points in 3D, which is a necessary criterion for a potential 3D absolute instability mechanism within the formalism of Briggs’ theory.