Claims of "scalar", "longitudinal" or "Tesla" electromagnetic waves often combine threedistinct phenomena: longitudinal electric-field components in anisotropic media, superluminalphase or group velocities, and an additional propagating scalar degree of freedom. Weseparate these questions and derive the relevant results from first principles. First, source-free Maxwell electrodynamics with tensor permittivity and permeability is reduced to itsplane-wave eigenvalue problem. The Gauss constraint is shown to make the displacementfield, rather than the electric field, transverse to the wave vector; an explicit anisotropicexample consequently has a predominantly longitudinal electric field and a phase velocityexceeding the vacuum light speed. The Poynting theorem is then derived for a dispersiveanisotropic medium, leading to the Brillouin energy density and energy velocity and showingwhy a superluminal phase velocity does not imply superluminal energy or signal propagation.Second, a genuine scalar degree of freedom is introduced through a Lorentz- and gauge-invariant dilaton-like action. All Euler—Lagrange equations are derived explicitly. Gauge,translation and Lorentz symmetries are treated with Noether’s theorem, yielding charge,energy—momentum and angular-momentum/boost conservation laws. Linearisation about astatic electric background gives the full mixed scalar—electromagnetic dispersion relation, itsnearly massless distinguished limit, and a scalar-associated branch which is purely longitudinalelectrically and has zero magnetic perturbation for propagation parallel to the backgroundfield. The same action gives a finite-electrode voltage-dependent capacitance; this predictionis derived using the Green function of the scalar equation. Existing scalar—photon andresonator searches already constrain the coupling strongly, making large effects in the minimalunscreened theory implausible. We finally formulate direct static and dynamical experimentaltests of the remaining longitudinal mode.