  We investigate the dripline mirror nuclei, $^{11}$Li and $^{11}$O, located on
the neutron and proton dripline, respectively. We calculate the lowest four
states, $3/2^-$, $1/2^+$, $3/2^+$ and $5/2^+$, built on double occupancy in the
nuclear $s_{1/2}$ and $p_{1/2}$ valence single-particle states. We use the
hyperspherical adiabatic expansion method to solve the three-body problem for a
frozen nuclear core surrounded by two identical nucleons. The four analogue
states in $^{11}$O are obtained with precisely the same interactions as used
for the four states in $^{11}$Li, except for addition of the Coulomb
interaction from the charge of the substituted valence protons. Surprisingly
the four energies deviate from each other only by less than a few hundred keV.
Any of them could then turn out to be the ground state, due to the uncertainty
related to the angular momentum and parity dependence of the three-body
potential. Still, our calculations marginally favor the $1/2^+$ state. The
structures of these four states in $^{11}$O deviate substantially from the
analogue states in the mirror, $^{11}$Li.
