One-dimensional nanomechanical resonators based on nanowires and nanotubes have emerged as promising candidates for mass sensors. When the resonator is clamped at one end and the atoms or molecules being measured land on the other end (which is free to vibrate), the resonance frequency of the device decreases by an amount that is proportional to the mass of the atoms or molecules. However, atoms and molecules can land at any position along the resonator, and many biomolecules have sizes that are comparable to the size of the resonator, so the relationship between the added mass and the frequency shift breaks down. Moreover, whereas resonators fabricated by top-down methods tend to vibrate in just one dimension because they are usually shaped like diving boards, perfectly axisymmetric one-dimensional nanoresonators can support flexural vibrations with the same amplitude and frequency in two dimensions. Here, we propose a new approach to mass sensing and stiffness spectroscopy based on the fact that the nanoresonator will enter a superposition state of two orthogonal vibrations with different frequencies when this symmetry is broken. Measuring these frequencies allows the mass, stiffness and azimuthal arrival direction of the adsorbate to be determined.