  Using an exact numerical solution and semiclassical analysis, we investigate
quantum oscillations (QOs) in a model of a bilayer system with an anisotropic
(elliptical) electron pocket in each plane. Key features of QO experiments in
the high temperature superconducting cuprate YBCO can be reproduced by such a
model, in particular the pattern of oscillation frequencies (which reflect
"magnetic breakdown" between the two pockets) and the polar and azimuthal
angular dependence of the oscillation amplitudes. However, the requisite
magnetic breakdown is possible only under the assumption that the horizontal
mirror plane symmetry is spontaneously broken and that the bilayer tunneling,
$t_\perp$, is substantially renormalized from its `bare' value. Under the
assumption that $t_\perp= \tilde{Z}t_\perp^{(0)}$, where $\tilde{Z}$ is a
measure of the quasiparticle weight, this suggests that $\tilde{Z} \lesssim
1/20$. Detailed comparisons with new YBa$_2$Cu$_3$O$_{6.58}$ QO data, taken
over a very broad range of magnetic field, confirm specific predictions made by
the breakdown scenario.
