Context. Major mergers are popularly considered too destructive to produce the relaxed regular structures and the morphological inner components (ICs) usually observed in lenticular (S0) galaxies. Aims: We aim to test if major mergers can produce remnants with realistic S0 morphologies. Methods: We have selected a sample of relaxed discy remnants resulting from the dissipative merger simulations of the GalMer database and derived their properties mimicking the typical conditions of current observational data. We have compared their global morphologies, visual components, and merger relics in mock photometric images with their real counterparts. Results: Only Ȉ1-2 Gyr after the full merger, we find that: 1) many remnants (67 major and 29 minor events) present relaxed structures and typical S0 or E/S0 morphologies, for a wide variety of orbits and even in gas-poor cases. 2) Contrary to popular expectations, most of them do not exhibit any morphological traces of their past merger origin under typical observing conditions and at distances as nearby as 30 Mpc. 3) The merger relics are more persistent in minor mergers than in major ones for similar relaxing time periods. 4) No major-merger S0-like remnant develops a significant bar. 5) Nearly 58% of the major-merger S0 remnants host visually detectable ICs, such as embedded inner discs, rings, pseudo-rings, inner spirals, nuclear bars, and compact sources, very frequent in real S0s too. 6) All remnants contain a lens or oval, identically ubiquitous in local S0s. 7) These lenses and ovals do not come from bar dilution in major-merger cases, but are associated with stellar halos or embedded inner discs instead (thick or thin). Conclusions: The relaxed morphologies, lenses, ovals, and other ICs of real S0s do not necessarily come from internal secular evolution, gas infall, or environmental mechanisms, as traditionally assumed, but they can result from major mergers as well. Full Tables A.2-A.4 are only available in electronic form at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/617/A113