Properties of the $\mathcal{M}$-Harmonic Conjugate Operator We define the $\mathcal{M}$-harmonic conjugate operator $K$ and prove that it is bounded on the nonisotropic Lipschitz space and on $\BMO$. Then we show $K$ maps Dini functions into the space of continuous functions on the unit sphere. We also prove the boundedness and compactness properties of $\mathcal{M}$-harmonic conjugate operator with $L^p$ symbol. Keywords:$\mathcal{M}$-harmonic conjugate operatorCategories:32A70, 47G10