This paper deals with the numerical evaluation of fundamental- and higher-mode solutions of the well-known K-eigenvalue problem in nuclear reactor physics using neutron transport theory. If the spatial domain has a plane of reflective symmetry, it is customary to find the fundamental K-mode by considering only the half-domain on one side of the plane for the calculation and applying a reflective boundary condition (RBC) on the plane of symmetry. Here, it is shown that the higher antisymmetric K-mode can also be evaluated in a similar way by applying what is called here the anti-RBC (ARBC) on the plane of symmetry. ARBC was implemented in computer codes based on the discrete ordinates method in Cartesian geometry for some sample problems and was found to work well. The implementation of ARBC in existing codes, although very easy, does not seem to be widely used or reported in the literature. For a one-dimensional homogeneous slab with isotropic scattering, the first antisymmetric K-mode found using ARBC is equivalent to the fundamental mode of a sphere, apart from a scaling factor for the total flux. An interesting result is that the fundamental mode of a sphere computed in this way does not contain the unphysical flux dip near the center, commonly obtained by the discrete ordinates codes in spherical geometry. Although not shown here, it appears that ARBC can be implemented in Monte Carlo codes also to find antisymmetric modes.