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They are broadly referred as linearly compatible, matching, and totally compatible structures. This paper gives a unified approach to these structures in the context of operads. Generalizing polarizations for polynomials in invariant theory to operads leads to linearly compatible operads. Partitioning polarizations into foliations gives matching operads which further yields total compatible operads under an invariance condition. For unary\/binary quadratic operads, linear compatibility and total compatibility are in Koszul dual, and the matching compatibilities are Koszul self-dual among themselves. For binary quadratic operads, these three compatible operads can be achieved by taking Manin products. 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