  The process of neutrinoless double electron capture ($0\nu$ECEC) is revisited
for those cases where the two participating atoms are nearly degenerate in
mass. The theoretical framework is the formalism of an oscillation of two atoms
with different total lepton number (and parity), one of which can be in an
excited state so that mass degeneracy is realized. In such a case and assuming
light Majorana neutrinos, the two atoms will be in a mixed configuration with
respect to the weak interaction. A resonant enhancement of transitions between
such pairs of atoms will occur, which could be detected by the subsequent
electromagnetic de-excitation of the excited state of the daughter atom and
nucleus. Available data of atomic masses, as well as nuclear and atomic
excitations are used to select the most likely candidates for such resonant
$0\nu$ECEC transitions. Assuming an effective mass for the Majorana neutrino of
1 eV, some half-lives are predicted to be as low as $10^{22}$ years in the
unitary limit. It is argued that, in order to obtain more accurate predictions
for the $0\nu$ECEC half-lives, precision mass measurements of the atoms
involved are necessary, which can readily be accomplished by today's high
precision Penning traps. Further advancements also require a better
understanding of high-lying excited states of the final nuclei (i.e. excitation
energy, angular momentum and parity) and the calculation of the nuclear matrix
elements.
