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In this paper, complete characterizations of left (or right) symmetric points for strong Birkhoff orthogonality in    $$B(\mathcal {H},\mathcal {K})$$    B  (  H  ,  K  )      and    $$K(\mathcal {H},\mathcal {K})$$    K  (  H  ,  K  )      are given, where    $$\mathcal {H},\mathcal {K}$$    H  ,  K      are complex Hilbert spaces and    $$B(\mathcal {H},\mathcal {K})$$    B  (  H  ,  K  )      (   $$K(\mathcal {H},\mathcal {K})$$    K  (  H  ,  K  )     ) is the space of all bounded linear (compact) operators from    $$\mathcal {H}$$   H     into    $$\mathcal {K}$$   K    .

On symmetry of strong Birkhoff orthogonality in $$B(\mathcal {H},\mathcal {K})$$ and $$K(\mathcal {H},\mathcal {K})$$