The PHENIX Collaboration has reported third-order harmonic oscillations of the source radius parameters when measuring the Hanbury-Brown-Twiss (HBT) correlation function for charged hadrons relative to the triangular flow angle. We explore possible origins of such third-order oscillations with a simple Gaussian emission function featuring both a triangular geometric deformation and triangular flow. Third-order oscillations of the HBT radii can arise from a purely geometric triangular deformation superimposed on an azimuthally symmetric radial flow, or from a radially symmetry spatial distribution which expands anisotropically with a triangular component in the flow velocity profile. In both cases the final particle momentum distribution features triangular flow. We show that the two alternatives can be distinguished experimentally through the phase of the azimuthal oscillations of the HBT radii relative to the triangular flow plane.