  The lowest-lying states of the Borromean nucleus $^{17}$Ne ($^{15}$O+$p$ +
$p$) and its mirror nucleus $^{17}$N ($^{15}$N+$n$ + $n$) are compared by using
the hyperspheric adiabatic expansion. Three-body resonances are computed by use
of the complex scaling method. The measured size of $^{15}$O and the low-lying
resonances of $^{16}$F ($^{15}$O+$p$) are first used as constraints to
determine both central and spin-dependent two-body interactions. The
interaction obtained reproduces relatively accurately both experimental
three-body spectra. The Thomas-Ehrman shifts, involving excitation energy
differences, are computed and found to be less than 3% of the total Coulomb
energy shift for all states.
