Abstract The theory of hybrid principles is presented together with the transformation rule for converting odd-parity approximations into even-parity approximations. This rule leads to a method which provides rigorous upper and lower bounds for the disadvantage factor for a reactor lattice cell. With these bounds very precise benchmarks have been constructed for representative lattices. It is found that a combination of even and odd-parity solutions for the neutron flux is much more efficient than solution based on either the even-parity or odd-parity. This is the basis of one synthesis scheme. In another synthesis method, a hybrid principle with trial functions for both the even- and odd-parity angular flux is used in conjunction with a classical principle with an odd-parity trial function. The synthesis process is efficient because the largest set of equations to be solved, i.e. the frame work, involves as few as one unknown per node of the finite element mesh. The effectiveness of the synthesis method is demonstrated for a thick shield problem.