A field theoretical approach, applied to the time-reversible system described by the ideal magnetohydrodynamic (MHD) equations, exposes the full generality of MHD spectral theory. MHD spectral theory, which classified waves and instabilities of static or stationary, usually axisymmetric or translationally symmetric configurations, actually governs the stability of flowing, (self-)gravitating, single fluid descriptions of nonlinear, time-dependent idealized plasmas, and this at any time during their nonlinear evolution. At the core of this theory is a self-adjoint operator G, discovered by Frieman and Rotenberg [Rev. Mod. Phys. 32, 898 (1960)] in its application to stationary (i.e., time-independent) plasma states. This Frieman-Rotenberg operator dictates the acceleration identified by a Lagrangian displacement field ξ, which connects two ideal MHD states in four-dimensional space-time that share initial conditions for density, entropy, and magnetic field. The governing equation reads d2ξdt2=G[ξ], as first noted by Cotsaftis and Newcomb [Nucl. Fusion, Suppl. Part 2, 447 and 451 (1962)]. The time derivatives at left are to be taken in the Lagrangian way, i.e., moving with the flow v. Physically realizable displacements must have finite energy, corresponding to being square integrable in the Hilbert space of displacements equipped with an inner product rule, for which the G operator is self-adjoint. The acceleration in the left-hand side features the Doppler-Coriolis operator v·∇, which is known to become an antisymmetric operator when restricting attention to stationary equilibria. Here, we present all derivations needed to get to these insights and connect results throughout the literature. A first illustration elucidates what can happen when self-gravity is incorporated and presents aspects that have been overlooked even in simple uniform media. Ideal MHD flows, as well as Euler flows, have essentially 6 + 1 wave types, where the 6 wave modes are organized through the essential spectrum of the G operator. These 6 modes are actually three pairs of modes, in which the Alfvén pair (a shear wave pair in hydro) sits comfortably at the middle. Each pair of modes consists of a leftgoing wave and a rightgoing wave, or equivalently stated, with one type traveling from past to future (forward) and the other type that goes from future to past (backward). The Alfvén pair is special, in its left-right categorization, while there is full degeneracy for the slow and fast pairs when reversing time and mirroring space. The Alfvén pair group speed diagram leads to the familiar Elsässer variables.