  We address the most general periodic travelling wave of the modified
Korteweg-de Vries (mKdV) equation written as a rational function of Jacobian
elliptic functions. By applying an algebraic method which relates the periodic
travelling waves and the squared periodic eigenfunctions of the Lax operators,
we characterize explicitly the location of eigenvalues in the periodic spectral
problem away from the imaginary axis. We show that Darboux transformations with
the periodic eigenfunctions remain in the class of the same periodic travelling
waves of the mKdV equation. In a general setting, there are exactly three
symmetric pairs of eigenvalues away from the imaginary axis, and we give a new
representation of the second non-periodic solution to the Lax equations for the
same eigenvalues. We show that Darboux transformations with the non-periodic
solutions to the Lax equations produce rogue waves on the periodic background,
which are either brought from infinity by propagating algebraic solitons or
formed in a finite region of the time-space plane.
