  We report an angle-resolved photoemission spectroscopy study of the
iron-based superconductor family, Ba$_{1-x}$Na$_x$Fe$_2$As$_2$. This system
harbors the recently discovered double-Q magnetic order appearing in a
reentrant C$_4$ phase deep within the underdoped regime of the phase diagram
that is otherwise dominated by the coupled nematic phase and collinear
antiferromagnetic order. From a detailed temperature-dependence study, we
identify the electronic response to the nematic phase in an orbital-dependent
band shift that strictly follows the rotational symmetry of the lattice and
disappears when the system restores C$_4$ symmetry in the low temperature
phase. In addition, we report the observation of a distinct electronic
reconstruction that cannot be explained by the known electronic orders in the
system.
