Orbits in the inner kpc of a triaxial galaxy are discussed, taking into account the effect of a central density concentration like a massive black hole, a dense stellar nucleus, or a de Vaucouleurs-type cusp. Since the box orbits that form the backbone of a triaxial galaxy pass arbitrarily close to the centre after long enough time, they will eventually be subjected to large-angle deflections by a central point mass, and the triaxiality of the inner part of the system will thereby be destroyed. A 108 M⊙ black hole is estimated to affect box orbits out to 1kpc in a Hubble time, while a similar influence of the observed (extended) nucleus in M31 reaches out to 500pc in the bulge. Regular box orbits persist, however, in systems with singular central density profiles such as implied by carrying the r1/4 law all the way to the centre. This result can be approximately understood in terms of the frequency ratio Ωr/Ωθ remaining close to the harmonic value of 2 for many orbits in the corresponding spherical potential. Finally, I discuss observable consequences of the box orbit scattering process and future work, and use the presence of isophote twists in the central parts of a number of elliptical galaxies to obtain approximate upper limits on the masses of the black holes these systems may contain.