It is common in the literature on electrodynamics and relativity theory that the transformation rules for the basic electrodynamical quantities are derived from the hypothesis that the relativity principle (RP) applies for Maxwell’s electrodynamics. As it will turn out from our analysis, these derivations raise several problems, and certain steps are logically questionable. This is, however, not our main concern in this paper. Even if these derivations were completely correct, they leave open the following questions: (1) Is (RP) a true law of nature for electrodynamical phenomena? (2) Are, at least, the transformation rules of the fundamental electrodynamical quantities, derived from (RP), true? (3) Is (RP) consistent with the laws of electrodynamics in a single inertial frame of reference? (4) Are, at least, the derived transformation rules consistent with the laws of electrodynamics in a single frame of reference? Obviously, (1) and (2) are empirical questions. In this paper, we will investigate problems (3) and (4). First we will give a general mathematical formulation of (RP) and covariance. It will be shown that covariance is not only not sufficient for the relativity principle, but it is not even necessary. In the second part, we will deal with the operational definitions of the fundamental electrodynamical quantities. As we will see, these semantic issues are not as trivial as one might think. In the third part of the paper, applying what J. S. Bell calls “Lorentzian pedagogy”—according to which the laws of physics in any one reference frame account for all physical phenomena—we will show that the transformation rules of the electrodynamical quantities are identical with the ones obtained by presuming the covariance of the coupled Maxwell–Lorentz equations, and that the covariance is indeed satisfied.