The fraction of star-forming to quiescent dwarf galaxies varies from almost infinity in the field to zero in the centers of rich galaxy clusters. What is causing this pronounced morphology-density relation? What do quiescent dwarf galaxies look like when studied in detail, and what conclusions can be drawn about their formation mechanism? Here we study a nearly magnitude-complete sample (–19 < Mr <–16 mag) of 121 Virgo cluster early types with deep near-infrared images from the SMAKCED project. We fit two-dimensional models with optional inner and outer components, as well as bar and lens components (in ~15% of the galaxies), to the galaxy images. While a single Sérsic function may approximate the overall galaxy structure, it does not entirely capture the light distribution of two-thirds of our galaxies, for which multicomponent models provide a better fit. This fraction of complex galaxies shows a strong dependence on luminosity, being larger for brighter objects. We analyze the global and component-specific photometric scaling relations of early-type dwarf galaxies and discuss similarities with bright early and late types. The dwarfs' global galaxy parameters show scaling relations that are similar to those of bright disk galaxies. The inner components are mostly fitted with Sérsic n values close to 1. At a given magnitude, they are systematically larger than the bulges of spirals, suggesting that they are not ordinary bulges. We argue that the multicomponent structures in early-type dwarfs are mostly a phenomenon inherent to the disks and may indeed stem from environmental processing. Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile, under program IDs 064.N-0288 and 085.B-0919.