Cavity optomechanics offers quantum cooling, quantum control and measurement
of small mechanical oscillators. However the optical backactions that underpin
quantum control can significantly disturb the oscillator modes: mechanical
frequencies are shifted by the optical spring effect and light-matter
hybridisation in strong coupling regimes; mechanical modes hybridise with each
other via the cavity mode. This is even more pertinent in the field of
levitated optomechanics, where optical trapping fully determines the mechanical
modes and their frequencies. Here, using the coherent-scattering (CS) set-up
that allowed quantum ground state cooling of a levitated nanoparticle, we show
that -- when trapping away from a node of the cavity standing wave -- the CS
field opposes optical spring shifts and mechanical mode hybridisation. At an
optimal cancellation point, independent of most experimental parameters, we
demonstrate experimentally that it is possible to strongly cavity cool and
control the {\em unperturbed} modes. Suppression of the cavity-induced mode
hybridisation in the x−y plane is quantified by measuring the
Sxy(ω) correlation spectra which are seen to always be
anti-correlated except at the cancellation point where they become
uncorrelated. The findings have implications for directional force sensing
using CS set-ups
