This paper presents a classification of all two-digits Haar systems and tiles on the two-dimension plane up to affine similarity. We obtain three cases, only two of them (rectangular and Dragon tile) are well known. In all of them we compute the H"older regularity in $L_2$ of corresponding Haar functions. The technique of calculating the H"older regularity is well known for univariate wavelets and it was recently expended into multivariate wavelets. These values also give us the information

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... t the rate of convergence of the corresponding subdivision algorithms and the rate of convergence of the cascade algorithm of the corresponding Haar decomposition.