From a spatial point of view, it very much looks like a changing electric field generates a changing magnetic field. And vice-versa. But this raises the problem of causality and locality.
Its general taken in physics the cause and effect are local. Hence, if we wave an electron, the changing electric field will generate a changing magnetic field. But then by locality, you can equally say, after the initial cause, that a changing magnetic field generates the changing electric field. Thus what causes which becomes ambiguous. Of course one could posit instead Liebniz pre-established harmony. (Actually, this view, which sounds incredible begins to make sense when there is no time, for then there can be no temporal cause amd effect).
This puzzle is resolved once we move into a spacetime view. Then the electric and magnetic spatial wave are combined into a single spacetime EM field strength. It is this that varies. And its only when we choose a split of spacetime into space and time that we obtain the electric and magnetic fields.