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Cavity optomechanics offers quantum cooling, quantum control and measurementof small mechanical oscillators. However the optical backactions that underpinquantum control can significantly disturb the oscillator modes: mechanicalfrequencies are shifted by the optical spring effect and light-matterhybridisation in strong coupling regimes; mechanical modes hybridise with eachother via the cavity mode. This is even more pertinent in the field oflevitated optomechanics, where optical trapping fully determines the mechanicalmodes and their frequencies. Here, using the coherent-scattering (CS) set-upthat allowed quantum ground state cooling of a levitated nanoparticle, we showthat -- when trapping away from a node of the cavity standing wave -- the CSfield opposes optical spring shifts and mechanical mode hybridisation. At anoptimal cancellation point, independent of most experimental parameters, wedemonstrate experimentally that it is possible to strongly cavity cool andcontrol the {\em unperturbed} modes. Suppression of the cavity-induced modehybridisation in the $x-y$ plane is quantified by measuring the$S_{xy}(\omega)$ correlation spectra which are seen to always beanti-correlated except at the cancellation point where they becomeuncorrelated. The findings have implications for directional force sensingusing CS set-ups.

Controlling mode orientations and frequencies in levitated cavity optomechanics