Here, we analyze the transverse-magnetic (TM) wave interaction with a pair of slabs, one being an epsilon-negative (ENG) layer in which the real part of permittivity is assumed to be negative while its permeability has positive real part, and the other being a mu-negative (MNG) layer that has the real part of its permeability negative but its permittivity has positive real part. Although wave interaction with each slab by itself has predictable features, we show that the juxtaposition and pairing of such ENG and MNG slabs may, under certain conditions, lead to some unusual features, such as resonance, complete tunneling, zero reflection and transparency. The field distributions inside and outside such paired slabs are analyzed, and the Poynting vector distributions in such structures are studied. Using equivalent transmission-line models, we obtain the conditions for the resonance, complete tunneling and transparency, and we justify and explain the field behavior in these resonant paired structures. Salient features of the tunneling conditions, such as the roles of material parameters, slab thicknesses, dissipation, and angle of incidence are discussed. The analogy and correspondence between the ENG-MNG pair and the pair of a slab of conventional material juxtaposed with a “double-negative” medium is also discussed. Finally, a conceptual idea for a potential application of such a “matched” lossless ENG-MNG pair in “ideal” image displacement and image reconstruction is proposed.