Control and manipulation of electric current and, especially, its degree of spin polarization (spin filtering) across single molecules are currently of great interest in the field of molecular spintronics. We explore one possible strategy based on the modification of nanojunction symmetry which can be realized, for example, by a mechanical strain. Such modification can activate new molecular orbitals which were inactive before due to their orbital mismatch with the electrodes conduction states. This can result in several important consequences such as (i) quantum interference effects appearing as Fano-like features in electron transmission and (ii) the change in molecular level hybridization with the electrodes states. We argue that the symmetry change can affect very differently two majority- and minority-spin conductances and thus alter significantly the resulting spin-filtering ratio as the junction symmetry is modified. We illustrate the idea for two basic molecular junctions: Ni/benzene/Ni (perpendicular vs tilted orientations) and Ni/Si chain/Ni (zigzag vs linear chains). In both cases, one highest occupied molecular orbital (HOMO) and one lowest unoccupied molecular orbital (LUMO) (out of HOMO and LUMO doublets) are important. In particular, their destructive interference with other orbitals leads to dramatic suppression of majority-spin conductance in low-symmetry configurations. For a minority-spin channel, on the contrary, the conductance is strongly enhanced when the symmetry is lowered due to an increase in hybridization strength. We believe that our results may offer a potential route for creating molecular devices with a large on-off ratio of spin polarization via quantum interference effects.