We establish reflection positivity of covariance operators, using properties of Dirichlet or Neumann boundary data. Osterwalder and Schrader isolated a basic positivity property in classical statistical mechanics which leads to the construction of a self adjoint contraction semigroup e x p ( -tH) acting on a Hilbert space ~. This yields the standard connection between quantum mechanics and path integrals. Verification of their condition in specific examples ultimately relies on a particular positivity condition for the inverse ( -AB + I ) -1 of a Laplace operator A B acting on L 2 (Ra). Here B denotes classical boundary conditions on the finite union of piecewise smooth hypersurfaces 1 ~ in R a. We show here that the positivity condition can be derived from monotonicity CD ~< CN of Dirichlet and of Neumann Laplace operators. Let Il denote a hyperplane in R a and let 0 = On denote reflection in Il. Let R_+ a denote the two connected components of Ra\II. Note that on L2(Rd), 0 is self adjoint and 02 =/. Let Il+ denote the orthogonal projection onto L2 (R_+a). DEFINITION 1. An operator C is 0 invariant if [0, C] = O. DEFINITION 2. A 0 invariant operator C is OS positive with respect to 0 il: II+OCII+ ~>0. (I) Remark. If [0, C] = 0, then (1) is equivalent to H_OCII_ >~ O.