Presented in this paper is an experimental and numerical study of direction-dependent asymmetry of resonant optical characteristics inherent to metal – multilayer-dielectric (MMD) nanostructure, which has much in common with the Tamm plasmonic configuration. We demonstrate that when a MMD structure is illuminated from opposite sides, there is a noticeable asymmetry of the forward/backward reflection resonances, contrasting with the strictly symmetrical transmission resonances indicating classical optical reciprocity. Comparative measurements were carried out on a metal film and a quasi-periodic dielectric structure, which are identical to the corresponding parameters of the MMD structure. Directional asymmetry of reflection and transmission is briefly discussed for a modified MMD structure with the Kerr nonlinearity.