  Algebraic structures with multiple copies of a given type of operations
interrelated by various compatibility conditions have long being studied in
mathematics and mathematical physics. 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. We first
generalize the process of polarization for polynomials in invariant theory to
the one for operads, leading to the general notion of linearly compatible
operads. Refining the polarization by partitioning it into foliations, we
obtain a notion of matching operads consolidating those appeared recently from
applications of regularity structures, and Volterra integral equations.
Distinguished among them is the leveled matching compatibility, which is unique
with respect to a fix ordering of the vertices in tree monomials. Equating all
matching compatibilities of a given operad leads to the totally compatible
operad of this operad.
  For unary/binary quadratic operads, the linear compatibility and the total
compatibility are in Koszul dual to each other, and there is a Koszul
self-duality among the matching compatibilities. In particular, the leveled
matching compatibility is Koszul self-dual. For binary quadratic operads, these
three compatible operads can also be obtained by taking Manin black and white
products. For finitely generated binary quadratic operad, we prove that the
Koszulity is preserved under taking each of the linear and total
compatibilities, as well as the leveled matching compatibility.
