It is commonly believed that a changing magnetic field generates a rotation of an electric field. This is not correct.
A changing magnetic field and the rotation of an electric field are equivalent expressions of the same underlying effect, namely a time dependent vector potential as in* $${\bf \nabla} \times {\bf E} \equiv {\bf \nabla} \times {\bf A}_t \equiv ({\bf \nabla} \times {\bf A})_t \equiv {\bf B}_t \,.$$
The origin of $${\bf \nabla} \times {\bf B} \equiv {\bf E}_t$$ is the wave equation, in its homogeneous form, $${\bf A}_{tt} = \Delta {\bf A} \,,$$ which in this form holds in the Lorenz gauge. Here indeed one could say that ${\bf A}_{tt}$ causes $\Delta {\bf A}$ and that brings the question 'How?'. All attempts to explain this have failed, among which the famous Aether theory.
You posed an excellent question!
*All uncompensated charges assumed stationary, that is $V_t=0$.